Mandelbrot set
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You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Generate and draw the Mandelbrot set.
Note that there are many algorithms to draw Mandelbrot set and there are many functions which generate it .
11l
<lang 11l>F mandelbrot(a)
R (0.<50).reduce(0i, (z, _) -> z * z + @a)
F step(start, step, iterations)
R (0 .< iterations).map(i -> @start + (i * @step))
V rows = (step(1, -0.05, 41).map(y -> (step(-2.0, 0.0315, 80).map(x -> (I abs(mandelbrot(x + 1i * @y)) < 2 {‘*’} E ‘ ’))))) print(rows.map(row -> row.join(‘’)).join("\n"))</lang>
ACL2
<lang Lisp>(defun abs-sq (z)
(+ (expt (realpart z) 2)
(expt (imagpart z) 2)))
(defun round-decimal (x places)
(/ (floor (* x (expt 10 places)) 1)
(expt 10 places)))
(defun round-complex (z places)
(complex (round-decimal (realpart z) places)
(round-decimal (imagpart z) places)))
(defun mandel-point-r (z c limit)
(declare (xargs :measure (nfix limit)))
(cond ((zp limit) 0)
((> (abs-sq z) 4) limit)
(t (mandel-point-r (+ (round-complex (* z z) 15) c)
c
(1- limit)))))
(defun mandel-point (z iters)
(- 5 (floor (mandel-point-r z z iters) (/ iters 5))))
(defun draw-mandel-row (im re cols width iters)
(declare (xargs :measure (nfix cols)))
(if (zp cols)
nil
(prog2$ (cw (coerce
(list
(case (mandel-point (complex re im)
iters)
(5 #\#)
(4 #\*)
(3 #\.)
(2 #\.)
(otherwise #\Space))) 'string))
(draw-mandel-row im
(+ re (/ (/ width 3)))
(1- cols)
width iters))))
(defun draw-mandel (im rows width height iters)
(if (zp rows)
nil
(progn$ (draw-mandel-row im -2 width width iters)
(cw "~%")
(draw-mandel (- im (/ (/ height 2)))
(1- rows)
width
height
iters))))
(defun draw-mandelbrot (width iters)
(let ((height (floor (* 1000 width) 3333)))
(draw-mandel 1 height width height iters)))</lang>
- Output:
> (draw-mandelbrot 60 100)
#
..
.####
. # .##.
##*###############.
#.##################
.######################.
######. #######################
##########.######################
##############################################
##########.######################
######. #######################
.######################.
#.##################
##*###############.
. # .##.
.####
..
Ada
mandelbrot.adb: <lang Ada>with Lumen.Binary; package body Mandelbrot is
function Create_Image (Width, Height : Natural) return Lumen.Image.Descriptor is
use type Lumen.Binary.Byte;
Result : Lumen.Image.Descriptor;
X0, Y0 : Float;
X, Y, Xtemp : Float;
Iteration : Float;
Max_Iteration : constant Float := 1000.0;
Color : Lumen.Binary.Byte;
begin
Result.Width := Width;
Result.Height := Height;
Result.Complete := True;
Result.Values := new Lumen.Image.Pixel_Matrix (1 .. Width, 1 .. Height);
for Screen_X in 1 .. Width loop
for Screen_Y in 1 .. Height loop
X0 := -2.5 + (3.5 / Float (Width) * Float (Screen_X));
Y0 := -1.0 + (2.0 / Float (Height) * Float (Screen_Y));
X := 0.0;
Y := 0.0;
Iteration := 0.0;
while X * X + Y * Y <= 4.0 and then Iteration < Max_Iteration loop
Xtemp := X * X - Y * Y + X0;
Y := 2.0 * X * Y + Y0;
X := Xtemp;
Iteration := Iteration + 1.0;
end loop;
if Iteration = Max_Iteration then
Color := 255;
else
Color := 0;
end if;
Result.Values (Screen_X, Screen_Y) := (R => Color, G => Color, B => Color, A => 0);
end loop;
end loop;
return Result;
end Create_Image;
end Mandelbrot;</lang>
mandelbrot.ads: <lang Ada>with Lumen.Image;
package Mandelbrot is
function Create_Image (Width, Height : Natural) return Lumen.Image.Descriptor;
end Mandelbrot;</lang>
test_mandelbrot.adb: <lang Ada>with System.Address_To_Access_Conversions; with Lumen.Window; with Lumen.Image; with Lumen.Events; with GL; with Mandelbrot;
procedure Test_Mandelbrot is
Program_End : exception;
Win : Lumen.Window.Handle; Image : Lumen.Image.Descriptor; Tx_Name : aliased GL.GLuint; Wide, High : Natural := 400;
-- Create a texture and bind a 2D image to it
procedure Create_Texture is
use GL;
package GLB is new System.Address_To_Access_Conversions (GLubyte);
IP : GLpointer;
begin -- Create_Texture
-- Allocate a texture name
glGenTextures (1, Tx_Name'Unchecked_Access);
-- Bind texture operations to the newly-created texture name
glBindTexture (GL_TEXTURE_2D, Tx_Name);
-- Select modulate to mix texture with color for shading
glTexEnvi (GL_TEXTURE_ENV, GL_TEXTURE_ENV_MODE, GL_MODULATE);
-- Wrap textures at both edges
glTexParameteri (GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_REPEAT);
glTexParameteri (GL_TEXTURE_2D, GL_TEXTURE_WRAP_T, GL_REPEAT);
-- How the texture behaves when minified and magnified
glTexParameteri (GL_TEXTURE_2D, GL_TEXTURE_MIN_FILTER, GL_NEAREST);
glTexParameteri (GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, GL_NEAREST);
-- Create a pointer to the image. This sort of horror show is going to
-- be disappearing once Lumen includes its own OpenGL bindings.
IP := GLB.To_Pointer (Image.Values.all'Address).all'Unchecked_Access;
-- Build our texture from the image we loaded earlier
glTexImage2D (GL_TEXTURE_2D, 0, GL_RGBA, GLsizei (Image.Width), GLsizei (Image.Height), 0,
GL_RGBA, GL_UNSIGNED_BYTE, IP);
end Create_Texture;
-- Set or reset the window view parameters
procedure Set_View (W, H : in Natural) is
use GL;
begin -- Set_View
GL.glEnable (GL.GL_TEXTURE_2D);
glClearColor (0.8, 0.8, 0.8, 1.0);
glMatrixMode (GL_PROJECTION);
glLoadIdentity;
glViewport (0, 0, GLsizei (W), GLsizei (H));
glOrtho (0.0, GLdouble (W), GLdouble (H), 0.0, -1.0, 1.0);
glMatrixMode (GL_MODELVIEW);
glLoadIdentity;
end Set_View;
-- Draw our scene
procedure Draw is
use GL;
begin -- Draw
-- clear the screen
glClear (GL_COLOR_BUFFER_BIT or GL_DEPTH_BUFFER_BIT);
GL.glBindTexture (GL.GL_TEXTURE_2D, Tx_Name);
-- fill with a single textured quad
glBegin (GL_QUADS);
begin
glTexCoord2f (1.0, 0.0);
glVertex2i (GLint (Wide), 0);
glTexCoord2f (0.0, 0.0);
glVertex2i (0, 0);
glTexCoord2f (0.0, 1.0);
glVertex2i (0, GLint (High));
glTexCoord2f (1.0, 1.0);
glVertex2i (GLint (Wide), GLint (High));
end;
glEnd;
-- flush rendering pipeline
glFlush;
-- Now show it
Lumen.Window.Swap (Win);
end Draw;
-- Simple event handler routine for keypresses and close-window events
procedure Quit_Handler (Event : in Lumen.Events.Event_Data) is
begin -- Quit_Handler
raise Program_End;
end Quit_Handler;
-- Simple event handler routine for Exposed events
procedure Expose_Handler (Event : in Lumen.Events.Event_Data) is
pragma Unreferenced (Event);
begin -- Expose_Handler
Draw;
end Expose_Handler;
-- Simple event handler routine for Resized events
procedure Resize_Handler (Event : in Lumen.Events.Event_Data) is
begin -- Resize_Handler
Wide := Event.Resize_Data.Width;
High := Event.Resize_Data.Height;
Set_View (Wide, High);
-- Image := Mandelbrot.Create_Image (Width => Wide, Height => High); -- Create_Texture;
Draw; end Resize_Handler;
begin
-- Create Lumen window, accepting most defaults; turn double buffering off
-- for simplicity
Lumen.Window.Create (Win => Win,
Name => "Mandelbrot fractal",
Width => Wide,
Height => High,
Events => (Lumen.Window.Want_Exposure => True,
Lumen.Window.Want_Key_Press => True,
others => False));
-- Set up the viewport and scene parameters Set_View (Wide, High);
-- Now create the texture and set up to use it Image := Mandelbrot.Create_Image (Width => Wide, Height => High); Create_Texture;
-- Enter the event loop
declare
use Lumen.Events;
begin
Select_Events (Win => Win,
Calls => (Key_Press => Quit_Handler'Unrestricted_Access,
Exposed => Expose_Handler'Unrestricted_Access,
Resized => Resize_Handler'Unrestricted_Access,
Close_Window => Quit_Handler'Unrestricted_Access,
others => No_Callback));
end;
exception
when Program_End =>
null;
end Test_Mandelbrot;</lang>
- Output:
ALGOL 68
Plot part of the Mandelbrot set as a pseudo-gif image.
<lang algol68> INT pix = 300, max iter = 256, REAL zoom = 0.33 / pix; [-pix : pix, -pix : pix] INT plane; COMPL ctr = 0.05 I 0.75 # center of set #;
- Compute the length of an orbit. #
PROC iterate = (COMPL z0) INT:
BEGIN COMPL z := 0, INT iter := 1;
WHILE (iter +:= 1) < max iter # not converged # AND ABS z < 2 # not diverged #
DO z := z * z + z0
OD;
iter
END;
- Compute set and find maximum orbit length. #
INT max col := 0; FOR x FROM -pix TO pix DO FOR y FROM -pix TO pix
DO COMPL z0 = ctr + (x * zoom) I (y * zoom);
IF (plane [x, y] := iterate (z0)) < max iter
THEN (plane [x, y] > max col | max col := plane [x, y])
FI
OD
OD;
- Make a plot. #
FILE plot; INT num pix = 2 * pix + 1; make device (plot, "gif", whole (num pix, 0) + "x" + whole (num pix, 0)); open (plot, "mandelbrot.gif", stand draw channel); FOR x FROM -pix TO pix DO FOR y FROM -pix TO pix
DO INT col = (plane [x, y] > max col | max col | plane [x, y]);
REAL c = sqrt (1- col / max col); # sqrt to enhance contrast #
draw colour (plot, c, c, c);
draw point (plot, (x + pix) / (num pix - 1), (y + pix) / (num pix - 1))
OD
OD; close (plot) </lang>
ALGOL W
Generates an ASCII Mandelbrot Set. Translated from the sample program in the Compiler/AST Interpreter task. <lang algolw>begin
% This is an integer ascii Mandelbrot generator, translated from the %
% Compiler/AST Interpreter Task's ASCII Mandelbrot Set example program %
integer leftEdge, rightEdge, topEdge, bottomEdge, xStep, yStep, maxIter;
leftEdge := -420;
rightEdge := 300;
topEdge := 300;
bottomEdge := -300;
xStep := 7;
yStep := 15;
maxIter := 200;
for y0 := topEdge step - yStep until bottomEdge do begin
for x0 := leftEdge step xStep until rightEdge do begin
integer x, y, i;
string(1) theChar;
y := 0;
x := 0;
theChar := " ";
i := 0;
while i < maxIter do begin
integer x_x, y_y;
x_x := (x * x) div 200;
y_y := (y * y) div 200;
if x_x + y_y > 800 then begin
theChar := code( decode( "0" ) + i );
if i > 9 then theChar := "@";
i := maxIter
end;
y := x * y div 100 + y0;
x := x_x - y_y + x0;
i := i + 1
end while_i_lt_maxIter ;
writeon( theChar );
end for_x0 ;
write();
end for_y0
end. </lang>
- Output:
1111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333222222222222222222222222222222222222222 1111111111111222222222233333333333333333333333344444456655544443333332222222222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 11111111111222222333333333333333333333334444444445555679@@@@7654444443333333222222222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98755544444433333332222222222222222222222222 1111111122223333333333333333333333344444444445556668@@@ @@@76555544444333333322222222222222222222222 1111111222233333333333333333333344444444455566667778@@ @987666555544433333333222222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@877779@5443333333322222222222222222222 1111112233333333333333333334444455555556679@ @@@ @@@@@@ 8544333333333222222222222222222 1111122333333333333333334445555555556666789@@@ @86554433333333322222222222222222 1111123333333333333444456666555556666778@@ @ @@87655443333333332222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65444333333332222222222222222 111133334444444455555668@@@@@@@@@@@@99@@@ @@765444333333333222222222222222 111133444444445555556778@@@ @@@@ @855444333333333222222222222222 11124444444455555668@99@@ @ @655444433333333322222222222222 11134555556666677789@@ @86655444433333333322222222222222 111 @@876555444433333333322222222222222 11134555556666677789@@ @86655444433333333322222222222222 11124444444455555668@99@@ @ @655444433333333322222222222222 111133444444445555556778@@@ @@@@ @855444333333333222222222222222 111133334444444455555668@@@@@@@@@@@@99@@@ @@765444333333333222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65444333333332222222222222222 1111123333333333333444456666555556666778@@ @ @@87655443333333332222222222222222 1111122333333333333333334445555555556666789@@@ @86554433333333322222222222222222 1111112233333333333333333334444455555556679@ @@@ @@@@@@ 8544333333333222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@877779@5443333333322222222222222222222 1111111222233333333333333333333344444444455566667778@@ @987666555544433333333222222222222222222222 1111111122223333333333333333333333344444444445556668@@@ @@@76555544444333333322222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98755544444433333332222222222222222222222222 11111111111222222333333333333333333333334444444445555679@@@@7654444443333333222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 1111111111111222222222233333333333333333333333344444456655544443333332222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333222222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211 1111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111
Amazing Hopper
Generate part of the Mandelbrot set as a binary-file. The generated file is binary, and the graph can be made with Matlab's "imshow" function.
Hopper-code & macros Version: <lang Amazing Hopper>
- !/usr/bin/hopper
- include <hopper.h>
main: ancho=500 alto=500 min real=-2 minReal=minreal min complex=-2 max real = 2 max complex = 2
mandel=0,{ancho,alto}nanarray(mandel)
i=1,a2=0,b2=0,a=0,b=0,t1=0,t2=0 {","}toksep tic(t1)
for(i=1,{i} le than (ancho),++i)
for(j=1,{j} le than (alto),++j)
{minreal} PLUS ( {max real}, minus (minReal), MUL BY ({i}minus(1)), DIV BY ({ancho}minus(1))),mov(a)
{min complex} PLUS ( {maxcomplex}, minus (mincomplex), MULBY ({j} minus(1)),DIVBY ({alto} minus (1)) ),mov(b)
a2=a,b2=b,brillo=1
k=100
__iterador__:
sqrdiff(a,b), plus (a2)
{2} mulby (a), mulby (b), plus (b2)
mov(b),mov(a)
if( {a,b} sqr add, gthan (4))
brillo=0, jmp(__out__)
endif
k--,jnz(__iterador__)
__out__:
[j,i]{brillo}put(mandel)
next
next toc(t1,t2) {"TIME = ",t2,"\n"}print {mandel,"mandel.dat"}save {0}return </lang>
Hopper-Basic high level language version: <lang Amazing Hopper>
- !/usr/bin/hopper
- include <hopper.h>
main:
ancho=500
alto=500
min real=-2
min complex=-2
max real = 2
max complex = 2
mandel=0,{ancho,alto}nanarray(mandel)
t1=0,t2=0
tic(t1)
- hl{
i=1
while(i<=ancho)
j=1
while(j<=alto)
a=min real + ((max real-min real) * (i-1)/(ancho-1))
b=min complex+((maxcomplex-mincomplex)*(j-1)/(alto-1))
a2=a
b2=b
k=0
brillo=1
while ( k<100 )
a1 = sqrdiff(a,b)
b1 = mul all(2,a,b)
a = a1 +a2
b = b1 + b2
if ( (sqradd(a,b)) > 4 )
brillo=0
break
end if
k+=1
wend
mandel[j,i]=brillo
j+=1
wend
i+=1
wend
}
toc(t1,t2)
{"TIME = ",t2,"\n"}print
{","}toksep
{mandel,"mandel.dat"}save
{0}return </lang>
Arturo
<lang rebol>inMandelbrot?: function [c][
z: to :complex [0 0]
do.times: 50 [
z: c + z*z
if 4 < abs z -> return false
]
return true
]
mandelbrot: function [settings][
y: 0
while [y < settings\height][
Y: settings\yStart + y * settings\yStep
x: 0
while [x < settings\width][
X: settings\xStart + x * settings\xStep
if? inMandelbrot? to :complex @[X Y] -> prints "*"
else -> prints " "
x: x + 1
]
print ""
y: y + 1
]
]
mandelbrot #[ yStart: 1.0 yStep: neg 0.05
xStart: neg 2.0 xStep: 0.0315
height: 40 width: 80 ]</lang>
- Output:
**
******
********
******
******** ** *
*** *****************
************************ ***
****************************
******************************
*******************************
************************************
* **********************************
** ***** * **********************************
*********** ************************************
************** ************************************
***************************************************
******************************************************
************************************************************************
******************************************************
***************************************************
************** ************************************
*********** ************************************
** ***** * **********************************
* **********************************
************************************
*******************************
******************************
****************************
************************ ***
*** *****************
******** ** *
******
********
******
**
AutoHotkey
<lang autohotkey>Max_Iteration := 256 Width := Height := 400
File := "MandelBrot." Width ".bmp" Progress, b2 w400 fs9, Creating Colours ... Gosub, CreateColours Gosub, CreateBitmap Progress, Off Gui, -Caption Gui, Margin, 0, 0 Gui, Add, Picture,, %File% Gui, Show,, MandelBrot Return
GuiClose: GuiEscape: ExitApp
- ---------------------------------------------------------------------------
CreateBitmap: ; create and save a 32bit bitmap file
- ---------------------------------------------------------------------------
; define header details HeaderBMP := 14 HeaderDIB := 40 DataOffset := HeaderBMP + HeaderDIB ImageSize := Width * Height * 4 ; 32bit FileSize := DataOffset + ImageSize Resolution := 3780 ; from mspaint
; create bitmap header
VarSetCapacity(IMAGE, FileSize, 0)
NumPut(Asc("B") , IMAGE, 0x00, "Char")
NumPut(Asc("M") , IMAGE, 0x01, "Char")
NumPut(FileSize , IMAGE, 0x02, "UInt")
NumPut(DataOffset , IMAGE, 0x0A, "UInt")
NumPut(HeaderDIB , IMAGE, 0x0E, "UInt")
NumPut(Width , IMAGE, 0x12, "UInt")
NumPut(Height , IMAGE, 0x16, "UInt")
NumPut(1 , IMAGE, 0x1A, "Short") ; Planes
NumPut(32 , IMAGE, 0x1C, "Short") ; Bits per Pixel
NumPut(ImageSize , IMAGE, 0x22, "UInt")
NumPut(Resolution , IMAGE, 0x26, "UInt")
NumPut(Resolution , IMAGE, 0x2A, "UInt")
; fill in Data Gosub, CreatePixels
; save Bitmap to file
FileDelete, %File%
Handle := DllCall("CreateFile", "Str", File, "UInt", 0x40000000
, "UInt", 0, "UInt", 0, "UInt", 2, "UInt", 0, "UInt", 0)
DllCall("WriteFile", "UInt", Handle, "UInt", &IMAGE, "UInt"
, FileSize, "UInt *", Bytes, "UInt", 0)
DllCall("CloseHandle", "UInt", Handle)
Return
- ---------------------------------------------------------------------------
CreatePixels: ; create pixels for [-2 < x < 1] [-1.5 < y < 1.5]
- ---------------------------------------------------------------------------
Loop, % Height // 2 + 1 {
yi := A_Index - 1
y0 := -1.5 + yi / Height * 3 ; range -1.5 .. +1.5
Progress, % 200*yi // Height, % "Current line: " 2*yi " / " Height
Loop, %Width% {
xi := A_Index - 1
x0 := -2 + xi / Width * 3 ; range -2 .. +1
Gosub, Mandelbrot
p1 := DataOffset + 4 * (Width * yi + xi)
NumPut(Colour, IMAGE, p1, "UInt")
p2 := DataOffset + 4 * (Width * (Height-yi) + xi)
NumPut(Colour, IMAGE, p2, "UInt")
}
}
Return
- ---------------------------------------------------------------------------
Mandelbrot: ; calculate a colour for each pixel
- ---------------------------------------------------------------------------
x := y := Iteration := 0
While, (x*x + y*y <= 4) And (Iteration < Max_Iteration) {
xtemp := x*x - y*y + x0
y := 2*x*y + y0
x := xtemp
Iteration++
}
Colour := Iteration = Max_Iteration ? 0 : Colour_%Iteration%
Return
- ---------------------------------------------------------------------------
CreateColours: ; borrowed from PureBasic example
- ---------------------------------------------------------------------------
Loop, 64 {
i4 := (i3 := (i2 := (i1 := A_Index - 1) + 64) + 64) + 64
Colour_%i1% := RGB(4*i1 + 128, 4*i1, 0)
Colour_%i2% := RGB(64, 255, 4*i1)
Colour_%i3% := RGB(64, 255 - 4*i1, 255)
Colour_%i4% := RGB(64, 0, 255 - 4*i1)
}
Return
- ---------------------------------------------------------------------------
RGB(r, g, b) { ; return 24bit color value
- ---------------------------------------------------------------------------
Return, (r&0xFF)<<16 | g<<8 | b
}</lang>
AWK
<lang AWK>BEGIN {
XSize=59; YSize=21;
MinIm=-1.0; MaxIm=1.0;MinRe=-2.0; MaxRe=1.0;
StepX=(MaxRe-MinRe)/XSize; StepY=(MaxIm-MinIm)/YSize;
for(y=0;y<YSize;y++)
{
Im=MinIm+StepY*y;
for(x=0;x<XSize;x++)
{
Re=MinRe+StepX*x; Zr=Re; Zi=Im;
for(n=0;n<30;n++)
{
a=Zr*Zr; b=Zi*Zi;
if(a+b>4.0) break;
Zi=2*Zr*Zi+Im; Zr=a-b+Re;
}
printf "%c",62-n;
}
print "";
}
exit;
}</lang>
- Output:
>>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<========== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======= >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= ><<<<;;;;;:::972456-567763 +9;;<<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< ><<<<;;;;;:::972456-567763 +9;;<<<<<<< >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<=======
B
This implements a 16bit fixed point arithmetic Mandelbrot set calculation.
<lang B>main() {
auto cx,cy,x,y,x2,y2; auto iter;
auto xmin,xmax,ymin,ymax,maxiter,dx,dy;
xmin = -8601; xmax = 2867; ymin = -4915; ymax = 4915;
maxiter = 32;
dx = (xmax-xmin)/79; dy = (ymax-ymin)/24;
cy=ymin;
while( cy<=ymax ) {
cx=xmin;
while( cx<=xmax ) {
x = 0;
y = 0;
x2 = 0;
y2 = 0;
iter=0;
while( iter<maxiter ) {
if( x2+y2>16384 ) break;
y = ((x*y)>>11)+cy;
x = x2-y2+cx;
x2 = (x*x)>>12;
y2 = (y*y)>>12;
iter++;
}
putchar(' '+iter);
cx =+ dx;
}
putchar(13);
putchar(10);
cy =+ dy;
}
return(0);
}</lang>
- Output:
!!!!!!!!!!!!!!!"""""""""""""####################################""""""""""""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$$%'+)%%%$$$$$#####"""""""""""
!!!!!!!!!!!"""""""#######################$$$$$$$$%%%&&(+,)++&%$$$$$$######""""""
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*5:/+('&%%$$$$$$#######"""
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''),@@@@@@@,'&%%%%%$$$$########
!!!!!!!"""####################$$$$$$$$%%%&'())((())*,@@@@@@/+))('&&&&)'%$$######
!!!!!!""###################$$$$$%%%%%%&&&'+.@@=/<@@@@@@@@@@@@@@@/++@..93%%$#####
!!!!!"################$$$%%%%%%%%%%&&&&'),+2@@@@@@@@@@@@@@@@@@@@@@@@@1(&&%$$####
!!!!"##########$$$$$%%&(-(''''''''''''(*,5@@@@@@@@@@@@@@@@@@@@@@@@@@@@+)-&%$$###
!!!!####$$$$$$$$%%%%%&'(*-@1.+.@-4+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4-(&%$$$##
!!!!#$$$$$$$$$%%%%%%'''++.6@@@@@@@@@8/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@3(%%$$$$#
!!!#$$$$$$$%&&&&''()/-5.5@@@@@@@@@@@@@>@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@?'&%%$$$$#
!!!(**+/+<523/80/46@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4+)'&&%%$$$$#
!!!#$$$$$$$%&&&&''().-2.@@@@@@@@@@@@@@?@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@'&%%$$$$#
!!!!#$$$$$$$$$%%%%%&'''/,.7@@@@@@@@@;/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@0'%%$$$$#
!!!!####$$$$$$$$%%%%%&'(*-:2.,/?-5+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4+(&%$$$##
!!!!"##########$$$$$%%&(-(''''(''''''((*,4@@@@@@@@@@@@@@@@@@@@@@@@@@@4+).&%$$###
!!!!!"################$$$%%%%%%%%%%&&&&')<,4@@@@@@@@@@@@@@@@@@@@@@@@@/('&%%$####
!!!!!!""##################$$$$$$%%%%%%&&&'*.@@@0@@@@@@@@@@@@@@@@1,,@//9)%%$#####
!!!!!!!"""####################$$$$$$$$%%%&(())((()**-@@@@@@/+)))'&&&')'%$$######
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''(,@@@@@@@+'&&%%%%%$$$########
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*7@0+('&%%%$$$$$#######"""
!!!!!!!!!!!"""""""######################$$$$$$$$$%%%&&(+-).*&%$$$$$$######""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$%%'3(%%%$$$$$######""""""""""
!!!!!!!!!!!!!!!""""""""""""#####################################""""""""""""""""
BASIC
AmigaBASIC
<lang amigabasic>SCREEN 1,320,200,5,1 WINDOW 2,"Mandelbrot",,0,1
maxIteration = 100 xmin = -2 xmax = 1 ymin = -1.5 ymax = 1.5 xs = 300 ys = 180 st = .01 ' use e.g. st = .05 for a coarser but faster picture
' and perhaps also lower maxIteration = 10 or so
xp = xs / (xmax - xmin) * st yp = ys / (ymax - ymin) * st
FOR x0 = xmin TO xmax STEP st
FOR y0 = ymin TO ymax STEP st
x = 0
y = 0
iteration = 0
WHILE (x * x + y * y <= (2 * 2) AND iteration < maxIteration)
xtemp = x * x - y * y + x0
y = 2 * x * y + y0
x = xtemp
iteration = iteration + 1
WEND
IF iteration <> maxIteration THEN
c = iteration
ELSE
c = 0
END IF
COLOR c MOD 32
AREA ((x0 - xmin) * xp / st, (y0 - ymin) * yp / st)
AREA STEP (xp, 0)
AREA STEP (0, yp)
AREA STEP (-xp, 0)
AREA STEP (0, -yp)
AREAFILL
NEXT
NEXT
' endless loop, use Run -> Stop from the menu to stop program WHILE (1) WEND</lang>
Applesoft BASIC
This version takes into account the Apple II's funky 280×192 6-color display, which has an effective resolution of only 140×192 in color.
<lang basic> 10 HGR2 20 XC = -0.5 : REM CENTER COORD X 30 YC = 0 : REM " " Y 40 S = 2 : REM SCALE 45 IT = 20 : REM ITERATIONS 50 XR = S * (280 / 192): REM TOTAL RANGE OF X 60 YR = S : REM " " " Y 70 X0 = XC - (XR/2) : REM MIN VALUE OF X 80 X1 = XC + (XR/2) : REM MAX " " X 90 Y0 = YC - (YR/2) : REM MIN " " Y 100 Y1 = YC + (YR/2) : REM MAX " " Y 110 XM = XR / 279 : REM SCALING FACTOR FOR X 120 YM = YR / 191 : REM " " " Y 130 FOR YI = 0 TO 3 : REM INTERLEAVE 140 FOR YS = 0+YI TO 188+YI STEP 4 : REM Y SCREEN COORDINATE 145 HCOLOR=3 : HPLOT 0,YS TO 279,YS 150 FOR XS = 0 TO 278 STEP 2 : REM X SCREEN COORDINATE 170 X = XS * XM + X0 : REM TRANSL SCREEN TO TRUE X 180 Y = YS * YM + Y0 : REM TRANSL SCREEN TO TRUE Y 190 ZX = 0 200 ZY = 0 210 XX = 0 220 YY = 0 230 FOR I = 0 TO IT 240 ZY = 2 * ZX * ZY + Y 250 ZX = XX - YY + X 260 XX = ZX * ZX 270 YY = ZY * ZY 280 C = IT-I 290 IF XX+YY >= 4 GOTO 301 300 NEXT I 301 IF C >= 8 THEN C = C - 8 : GOTO 301 310 HCOLOR = C : HPLOT XS, YS TO XS+1, YS 320 NEXT XS 330 NEXT YS 340 NEXT YI </lang>
By making the following modifications, the same code will render the Mandelbrot set in monochrome at full 280×192 resolution.
<lang basic> 150 FOR XS = 0 TO 279 301 C = (C - INT(C/2)*2)*3 310 HCOLOR = C: HPLOT XS, YS </lang>
ARM BBC BASIC
The ARM development second processor for the BBC series of microcomputers came with ARM BBC Basic V. In version 1.0 this included a built-in MANDEL function that uses D% (depth) to update C% (colour) at given coordinates x and y. Presumably this was for benchmarking/demo purposes; it was removed from later versions. This can be run in BeebEm. Select BBC model as Master 128 with ARM Second Processor. Load disc armdisc3.adl and switch to ADFS. At the prompt load ARM Basic by running the AB command.
<lang bbcbasic> 10MODE5:VDU5 20D%=100 : REM adjust for speed/precision 30FORX%=0 TO 1279 STEP8 40FORY%=0 TO 1023 STEP4 50MANDEL (Y%-512)/256, (X%-640)/256 51REM (not sure why X and Y need to be swapped to correct orientation) 60GCOL0,C% 70PLOT69,X%,Y% 80NEXT 90NEXT</lang> File:Mandelbrot armbasic.png
BASIC256
<lang basic-256>fastgraphics
graphsize 384,384 refresh kt = 319 : m = 4.0 xmin = -2.1 : xmax = 0.6 : ymin = -1.35 : ymax = 1.35 dx = (xmax - xmin) / graphwidth : dy = (ymax - ymin) / graphheight
for x = 0 to graphwidth jx = xmin + x * dx for y = 0 to graphheight jy = ymin + y * dy k = 0 : wx = 0.0 : wy = 0.0 do tx = wx * wx - wy * wy + jx ty = 2.0 * wx * wy + jy wx = tx wy = ty r = wx * wx + wy * wy k = k + 1 until r > m or k > kt
if k > kt then color black else if k < 16 then color k * 8, k * 8, 128 + k * 4 if k >= 16 and k < 64 then color 128 + k - 16, 128 + k - 16, 192 + k - 16 if k >= 64 then color kt - k, 128 + (kt - k) / 2, kt - k end if plot x, y next y refresh next x imgsave "Mandelbrot_BASIC-256.png", "PNG" </lang>
- Image generated by the script:
BBC BASIC
<lang bbcbasic> sizex% = 300 : sizey% = 300
maxiter% = 128
VDU 23,22,sizex%;sizey%;8,8,16,128
ORIGIN 0,sizey%
GCOL 1
FOR X% = 0 TO 2*sizex%-2 STEP 2
xi = X%/200 - 2
FOR Y% = 0 TO sizey%-2 STEP 2
yi = Y% / 200
x = 0
y = 0
FOR I% = 1 TO maxiter%
IF x*x+y*y > 4 EXIT FOR
xt = xi + x*x-y*y
y = yi + 2*x*y
x = xt
NEXT
IF I%>maxiter% I%=0
COLOUR 1,I%*15,I%*8,0
PLOT X%,Y% : PLOT X%,-Y%
NEXT
NEXT X%</lang>
Commander X16 BASIC
<lang X16BASIC> 10 CLS 20 SCREEN $80 30 FOR X=1 TO 199: 40 FOR Y=1 TO 99: 50 LET I=0 60 LET CX=(X-100)/50 70 LET CY=(Y-100)/50 80 LET VX=0 90 LET VY=0 100 REM START OF THE CALCULATION LOOP 110 LET I=I+1 120 LET X2 = VX*VX 130 LET Y2 = VY*VY 140 LET VY = CY + (VX+VX)*VY 150 LET VX = CX + X2-Y2 160 IF I<32 AND (X2+Y2)<4 THEN GOTO 100 170 LET YR = 199-Y 180 PSET X,Y,I 190 PSET X,YR,I 200 :NEXT:NEXT </lang>
Commodore BASIC
The standard version 2.0 of BASIC that came with the VIC-20 and C-64 had no built-in support for graphics (though you could use POKE statements to put values in the VIC chip's registers and the RAM being used for video). However, Commodore sold add-on cartridges that provided such support, which was also included in later versions of BASIC (3.5 on the C-16 and Plus/4, 7.0 on the C-128). In all of these cases the programs are very slow; rendering the Mandelbrot set even with the escape threshold set to just 20 iterations takes hours. The fastest is the VIC-20, because it has only a 160x160-pixel bitmap display.
VIC-20 with Super Expander cartridge
Runs in about 90 minutes.
<lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 COLOR 1,3,0,0:GRAPHIC 2:TH=20 120 FOR PY=0 TO 80 130 : SY=INT(PY*6.4) 140 : IM=1.12-PY/160*2.24 150 : FOR PX=0 TO 160 160 : SX=INT(PX*6.4) 170 : RE=PX/160*2.47-2 180 : X=0:Y=0:IT=0 190 : XT=X*X-Y*Y+RE 200 : Y = 2*X*Y+IM 210 : X=XT 220 : IT=IT+1 230 : IF(X*X+Y*Y<=4)AND(IT<TH) THEN 190 240 : IF IT<TH THEN 270 250 : DRAW 1,SX,SY TO SX,SY 260 : DRAW 1,SX,1024-SY TO SX,1024-SY 270 : NEXT PX 280 NEXT PY 290 GET K$:IF K$="" THEN 290 300 GRAPHIC 4</lang>
C-64 with Super Expander 64
Runs in about 4.5 hours.
<lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 COLOR 6,0,0,0,6:GRAPHIC 2,1 120 FOR PY=0 TO 100 140 : IM=1.12-PY/200*2.24 150 : FOR PX=0 TO 319 170 : RE=PX/320*2.47-2 180 : X=0:Y=0:IT=0 190 : XT=X*X-Y*Y+RE 200 : Y = 2*X*Y+IM 210 : X=XT 220 : IT=IT+1 230 : IF(X*X+Y*Y<=4)AND(IT<TH) THEN 190 240 : IF IT<TH THEN 270 250 : DRAW 1,PX,PY 260 : DRAW 1,PX,200-PY 270 : NEXT PX 280 NEXT PY 290 GET K$:IF K$="" THEN 290 300 GRAPHIC 0</lang>
Commodore-16 / 116 / Plus/4
Despite the faster clock on the TED systems compared to the C-64, this takes almost six hours to run. <lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 COLOR 0,2:COLOR 1,1:GRAPHIC 1,1 120 FOR PY=0 TO 100 130 : IM = 1.12-PY*2.24/200 140 : FOR PX=0 TO 319 150 : RE=PX/320*2.47-2 160 : X=0:Y=0:IT=0 170 : DO WHILE (X*X+Y*Y<=4) AND (IT<TH) 180 : XT = X*X-Y*Y+RE 190 : Y = 2*X*Y+IM 200 : X = XT 210 : IT = IT +1 220 : LOOP 230 : IF IT=TH THEN DRAW 1,PX,PY:DRAW 1,PX,199-PY 240 : NEXT PX 250 NEXT PY 260 GETKEY K$ 270 GRAPHIC 0</lang>
Commodore 128 (40-column display)
With the switch to FAST (2MHz) mode, this runs in about 2.5 hours, but you get to stare at a blank screen until it's done rendering. Without that switch it takes about 5.5 hours, splitting the difference between the Super Expander 64 and Plus/4 versions.
<lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 COLOR 0,12:GRAPHIC 1,1:COLOR 1,1:GRAPHIC0:FAST 120 FOR PY=0 TO 100 130 : IM = 1.12-PY*2.24/200 140 : FOR PX=0 TO 319 150 : RE=PX/320*2.47-2 160 : X=0:Y=0:IT=0 170 : DO WHILE (X*X+Y*Y<=4) AND (IT<TH) 180 : XT = X*X-Y*Y+RE 190 : Y = 2*X*Y+IM 200 : X = XT 210 : IT = IT +1 220 : LOOP 230 : IF IT=TH THEN DRAW 1,PX,PY:DRAW 1,PX,199-PY 240 : NEXT PX 250 NEXT PY 260 SLOW:GRAPHIC 1 270 GETKEY K$ 280 GRAPHIC 0</lang>
Commodore 128 (80-column display)
This uses BASIC 8 to create a 640x200 render on the C-128's 80-column display. The doubled resolution comes with a commensurate increase in run time; this takes about 5h20m using FAST 2MHz mode.
<lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 @MODE,0:@COLOR,15,0,0:@SCREEN,0,0:@CLEAR,0:FAST 120 FOR PY=0 TO 100 130 : IM = 1.12-PY*2.24/200 140 : FOR PX=0 TO 639
150 : RE=PX/640*2.47-2 160 : X=0:Y=0:IT=0 170 : DO WHILE (X*X+Y*Y<=4) AND (IT<20) 180 : XT = X*X-Y*Y+RE 190 : Y = 2*X*Y+IM 200 : X = XT 210 : IT = IT +1 220 : LOOP 230 : IF IT=20 THEN @DOT,PX,PY,0:@DOT,PX,199-PY,0 240 : NEXT PX 250 NEXT PY 260 GETKEY K$ 270 @TEXT</lang>
- Output:
Screen shots of all of the above running in the VICE emulator can be found here.
Commodore PET
Here's a version using mostly ASCII and some PETSCII (could probably improve the tile set for PETSCII) inspired by the Perl solution. Designed for a PET with an 80-column display. <lang basic>100 TH=20:REM ESCAPE THRESHOLD IN ITERATIONS 110 CH$=" .:-=+*#@"+CHR$(255) 120 FOR Y=1 TO -1 STEP -0.08 130 : FOR X = -2 TO 0.5 STEP 0.0315 140 : GOSUB 200:IF I=20 THEN PRINT CHR$(18);" ";CHR$(146);:GOTO 160 150 : PRINT MID$(CH$,I/2+1,1); 160 : NEXT X:IF PEEK(198) THEN PRINT 170 NEXT Y 180 GET K$:IF K$=""THEN 180 190 END 200 CX=X:CY=Y:ZX=X:ZY=Y 210 FOR I=1 TO TH 220 : TX=ZX*ZX-ZY*ZY 230 : TY=ZX*ZY*2 240 : ZX = TX + CX 250 : ZY = TY + CY 260 : IF ZX*ZX+ZY*ZY > 4 THEN 280 270 NEXT I 280 IF I>TH THEN I=TH 290 RETURN</lang>
- Output:
VICE screenshot here.
DEC BASIC-PLUS
Works under RSTS/E v7.0 on the simh PDP-11 emulator. For installation procedures for RSTS/E, see here. <lang>10 X1=59\Y1=21 20 I1=-1.0\I2=1.0\R1=-2.0\R2=1.0 30 S1=(R2-R1)/X1\S2=(I2-I1)/Y1 40 FOR Y=0 TO Y1 50 I3=I1+S2*Y 60 FOR X=0 TO X1 70 R3=R1+S1*X\Z1=R3\Z2=I3 80 FOR N=0 TO 30 90 A=Z1*Z1\B=Z2*Z2 100 IF A+B>4.0 THEN GOTO 130 110 Z2=2*Z1*Z2+I3\Z1=A-B+R3 120 NEXT N 130 PRINT STRING$(1%,62%-N); 140 NEXT X 150 PRINT 160 NEXT Y 170 END </lang>
- Output:
>>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<=========== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== ><<<<;;;;;:::972456-567763 +9;;<<<<<<<= ><;;;;;;::::9875& .3 *9;;;<<<<<<= >;;;;;;::997564' ' 8:;;;<<<<<<= >::988897735/ &89:;;;<<<<<<= >::988897735/ &89:;;;<<<<<<= >;;;;;;::997564' ' 8:;;;<<<<<<= ><;;;;;;::::9875& .3 *9;;;<<<<<<= ><<<<;;;;;:::972456-567763 +9;;<<<<<<<= >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== >>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<===========
FreeBASIC
<lang freebasic>#define pix 1./120
- define zero_x 320
- define zero_y 240
- define maxiter 250
screen 12
type complex
r as double i as double
end type
operator + (x as complex, y as complex) as complex
dim as complex ret ret.r = x.r + y.r ret.i = x.i + y.i return ret
end operator
operator * (x as complex, y as complex) as complex
dim as complex ret ret.r = x.r*y.r - x.i*y.i ret.i = x.r*y.i + x.i*y.r return ret
end operator
operator abs ( x as complex ) as double
return sqr(x.r*x.r + x.i*x.i)
end operator
dim as complex c, z dim as integer x, y, iter
for x=0 to 639
for y=0 to 240
c.r = (x-zero_x)*pix
c.i = (y-zero_y)*pix
z.r = 0.0
z.i = 0.0
for iter=0 to maxiter
z = z*z + c
if abs(z)>2 then
pset(x,y),iter mod 16
pset(x,480-y),iter mod 16
goto cont
end if
next iter
pset(x,y),1
pset(x,480-y),1
cont:
next y
next x
while inkey="" wend end</lang>
GW-BASIC
<lang gwbasic>10 SCALE# = 1/60 : ZEROX = 160 20 ZEROY = 100 : MAXIT = 32 30 SCREEN 1 40 FOR X = 0 TO 2*ZEROX - 1 50 CR# = (X-ZEROX)*SCALE# 60 FOR Y = 0 TO ZEROY 70 CI# = (ZEROY-Y)*SCALE# 80 ZR# = 0 90 ZI# = 0 100 FOR I = 1 TO MAXIT 110 BR# = CR# + ZR#*ZR# - ZI#*ZI# 120 ZI# = CI# + 2*ZR#*ZI# 130 ZR# = BR# 140 IF ZR#*ZR# + ZI#*ZI# > 4 THEN GOTO 170 150 NEXT I 160 GOTO 190 170 PSET (X, Y), 1 + (I MOD 3) 180 PSET (X, 2*ZEROY-Y), 1+(I MOD 3) 190 NEXT Y 200 NEXT X</lang>
Liberty BASIC
Any words of description go outside of lang tags. <lang lb>nomainwin
WindowWidth =440 WindowHeight =460
open "Mandelbrot Set" for graphics_nsb_nf as #w
- w "trapclose [quit]"
- w "down"
for x0 = -2 to 1 step .0033
for y0 = -1.5 to 1.5 step .0075
x = 0
y = 0
iteration = 0
maxIteration = 255
while ( ( x *x +y *y) <=4) and ( iteration <maxIteration)
xtemp =x *x -y *y +x0
y =2 *x *y +y0
x = xtemp
iteration = iteration + 1
wend
if iteration <>maxIteration then
c =iteration
else
c =0
end if
call pSet x0, y0, c
scan
next
next
- w "flush"
wait
sub pSet x, y, c
xScreen = 10 +( x +2) /3 *400
yScreen = 10 +( y +1.5) /3 *400
if c =0 then
col$ ="red"
else
if c mod 2 =1 then col$ ="lightgray" else col$ ="white"
end if
#w "color "; col$
#w "set "; xScreen; " "; yScreen
end sub
[quit] close #w end </lang>
Locomotive Basic
This program is meant for use in CPCBasic specifically, where it draws a 16-color 640x400 image in less than a minute. (Real CPC hardware would take far longer than that and has lower resolution.) <lang locobasic>1 MODE 3 ' Note the CPCBasic-only screen mode! 2 FOR xp = 0 TO 639 3 FOR yp = 0 TO 399 4 x = 0 : y = 0 5 x0 = xp / 213 - 2 : y0 = yp / 200 - 1 6 iteration = 0 7 maxIteration = 100 8 WHILE (x * x + y * y <= (2 * 2) AND iteration < maxIteration) 9 xtemp = x * x - y * y + x0 10 y = 2 * x * y + y0 11 x = xtemp 12 iteration = iteration + 1 13 WEND 14 IF iteration <> maxIteration THEN c = iteration ELSE c = 0 15 PLOT xp, yp, c MOD 16 16 NEXT 17 NEXT</lang>
Microsoft Small Basic
<lang Small BASIC> GraphicsWindow.Show() size = 500 half = 250 GraphicsWindow.Width = size * 1.5 GraphicsWindow.Height = size GraphicsWindow.Title = "Mandelbrot" For px = 1 To size * 1.5
x_0 = px/half - 2
For py = 1 To size
y_0 = py/half - 1
x = x_0
y = y_0
i = 0
While(c <= 2 AND i<100)
x_1 = Math.Power(x, 2) - Math.Power(y, 2) + x_0
y_1 = 2 * x * y + y_0
c = Math.Power(Math.Power(x_1, 2) + Math.Power(y_1, 2), 0.5)
x = x_1
y = y_1
i = i + 1
EndWhile
If i < 99 Then
GraphicsWindow.SetPixel(px, py, GraphicsWindow.GetColorFromRGB((255/25)*i, (255/25)*i, (255/5)*i))
Else
GraphicsWindow.SetPixel(px, py, "black")
EndIf
c=0
EndFor
EndFor </lang>
Microsoft Super Extended Color BASIC (Tandy Color Computer 3)
<lang COCO3BASIC> 1 REM MANDELBROT SET - TANDY COCO 3 2 POKE 65497,1 10 HSCREEN 2 20 HCLS 30 X1=319:Y1=191 40 I1=-1.0:I2=1.0:R1=-2:R2=1.0 50 S1=(R2-R1)/X1:S2=(I2-I1)/Y1 60 FOR Y=0 TO Y1 70 I3=I1+S2*Y 80 FOR X=0 TO X1 90 R3=R1+S1*X:Z1=R3:Z2=I3 100 FOR N=0 TO 30 110 A=Z1*Z1:B=Z2*Z2 120 IF A+B>4.0 GOTO 150 130 Z2=2*Z1*Z2+I3:Z1=A-B+R3 140 NEXT N 150 HSET(X,Y,N-16*INT(N/16)) 160 NEXT X 170 NEXT Y 180 GOTO 180 </lang>
Nascom BASIC
In fact, it is based on the ZX81 version, but some optimizations are done to shorten the execution time (though they lengthen the code):
- The variables
X2andY2forXA*XAandYA*YArespectively are introduced in order to calculate the squares only once. - The value of
Xdoes not depend onJ, so it is calculated before theFOR Jloop. - The symmetry of the shape is taken into account in order to calculate symmetric values only once. It is the most significant optimization. The case
X=0(corresponding to the line 22 on the screen grid) is treated separately.
Only a fragment of the shape is drawn because of low resolution of block graphics. Like in the ZX81 version, you can adjust the constants in lines 40 and 70 to zoom in on a particular area, if you like.
<lang basic> 10 REM Mandelbrot set 20 CLS 30 FOR I=0 TO 95 40 LET X=(I-78)/48 50 REM ** When X<>0 60 FOR J=1 TO 22 70 LET Y=(22-J)/30 80 LET XA=0:X2=0 90 LET YA=0:Y2=0 100 LET ITER=0 110 LET XTEMP=X2-Y2+X 120 LET YA=2*XA*YA+Y:Y2=YA*YA 130 LET XA=XTEMP:X2=XA*XA 140 LET ITER=ITER+1 150 IF X2+Y2<=4 AND ITER<200 THEN 110 160 IF ITER=200 THEN SET(I,J):SET(I,44-J) 170 NEXT J 180 REM ** When X=0 190 LET XA=0:X2=0 200 LET ITER=0 210 LET XA=X2+X:X2=XA*XA 220 LET ITER=ITER+1 230 IF X2<=4 AND ITER<200 THEN 210 240 IF ITER=200 THEN SET(I,22) 250 NEXT I 290 REM ** Set up machine code INKEY$ command 300 IF PEEK(1)<>0 THEN RESTORE 510 310 DOKE 4100,3328:FOR A=3328 TO 3342 STEP 2 320 READ B:DOKE A,B:NEXT A 400 SCREEN 1,15 410 PRINT "Hit any key to exit."; 420 A=USR(0):IF A<0 THEN 420 430 CLS 440 END 490 REM ** Data for machine code INKEY$ 500 DATA 25055,1080,-53,536,-20665,3370,-5664,0 510 DATA 27085,14336,-13564,6399,18178,10927 520 DATA -8179,233 </lang>
OS/8 BASIC
Works under BASIC on a PDP-8 running OS/8. Various emulators exist including simh's PDP-8 emulator and the PDP-8/E Simulator for Classic Macintosh and OS X. <lang qbasic>10 X1=59\Y1=21 20 I1=-1.0\I2=1.0\R1=-2.0\R2=1.0 30 S1=(R2-R1)/X1\S2=(I2-I1)/Y1 40 FOR Y=0 TO Y1 50 I3=I1+S2*Y 60 FOR X=0 TO X1 70 R3=R1+S1*X\Z1=R3\Z2=I3 80 FOR N=0 TO 30 90 A=Z1*Z1\B=Z2*Z2 100 IF A+B>4.0 GOTO 130 110 Z2=2*Z1*Z2+I3\Z1=A-B+R3 120 NEXT N 130 PRINT CHR$(62-N); 140 NEXT X 150 PRINT 160 NEXT Y 170 END</lang>
- Output:
>>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<=========== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== ><<<<;;;;;:::972456-567763 +9;;<<<<<<<= ><;;;;;;::::9875& .3 *9;;;<<<<<<= >;;;;;;::997564' ' 8:;;;<<<<<<= >::988897735/ &89:;;;<<<<<<= >::988897735/ &89:;;;<<<<<<= >;;;;;;::997564' ' 8:;;;<<<<<<= ><;;;;;;::::9875& .3 *9;;;<<<<<<= ><<<<;;;;;:::972456-567763 +9;;<<<<<<<= >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== >>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<===========
PureBasic
PureBasic forum: discussion <lang PureBasic>EnableExplicit
- Window1 = 0
- Image1 = 0
- ImgGadget = 0
- max_iteration = 64
- width = 800
- height = 600
Define.d x0 ,y0 ,xtemp ,cr, ci Define.i i, n, x, y ,Event ,color
Dim Color.l (255) For n = 0 To 63
Color( 0 + n ) = RGB( n*4+128, 4 * n, 0 ) Color( 64 + n ) = RGB( 64, 255, 4 * n ) Color( 128 + n ) = RGB( 64, 255 - 4 * n , 255 ) Color( 192 + n ) = RGB( 64, 0, 255 - 4 * n )
Next
If OpenWindow(#Window1, 0, 0, #width, #height, "'Mandelbrot set' PureBasic Example", #PB_Window_SystemMenu )
If CreateImage(#Image1, #width, #height)
ImageGadget(#ImgGadget, 0, 0, #width, #height, ImageID(#Image1))
For y.i = 1 To #height -1
StartDrawing(ImageOutput(#Image1))
For x.i = 1 To #width -1
x0 = 0
y0 = 0;
cr = (x / #width)*2.5 -2
ci = (y / #height)*2.5 -1.25
i = 0
While (x0*x0 + y0*y0 <= 4.0) And i < #max_iteration
i +1
xtemp = x0*x0 - y0*y0 + cr
y0 = 2*x0*y0 + ci
x0 = xtemp
Wend
If i >= #max_iteration
Plot(x, y, 0 )
Else
Plot(x, y, Color(i & 255))
EndIf
Next
StopDrawing()
SetGadgetState(#ImgGadget, ImageID(#Image1))
Repeat
Event = WindowEvent()
If Event = #PB_Event_CloseWindow
End
EndIf
Until Event = 0
Next
EndIf
Repeat
Event = WaitWindowEvent()
Until Event = #PB_Event_CloseWindow
EndIf</lang>Example:
QBasic
This is almost exactly the same as the pseudocode from the Wikipedia entry's "For programmers" section (which it's closely based on, of course). The image generated is very blocky ("low-res") due to the selected video mode, but it's fairly accurate.
<lang qbasic>SCREEN 13 WINDOW (-2, 1.5)-(2, -1.5) FOR x0 = -2 TO 2 STEP .01
FOR y0 = -1.5 TO 1.5 STEP .01
x = 0
y = 0
iteration = 0
maxIteration = 223
WHILE (x * x + y * y <= (2 * 2) AND iteration < maxIteration)
xtemp = x * x - y * y + x0
y = 2 * x * y + y0
x = xtemp
iteration = iteration + 1
WEND
IF iteration <> maxIteration THEN
c = iteration
ELSE
c = 0
END IF
PSET (x0, y0), c + 32 NEXT
NEXT</lang>
Quite BASIC
<lang Quite BASIC> 1000 REM Mandelbrot Set Project 1010 REM Quite BASIC Math Project 1015 REM 'http://www.quitebasic.com/prj/math/mandelbrot/ 1020 REM ------------------------ 1030 CLS 1040 PRINT "This program plots a graphical representation of the famous Mandelbrot set. It takes a while to finish so have patience and don't have too high expectations; the graphics resolution is not very high on our canvas." 2000 REM Initialize the color palette 2010 GOSUB 3000 2020 REM L is the maximum iterations to try 2030 LET L = 100 2040 FOR I = 0 TO 100 2050 FOR J = 0 TO 100 2060 REM Map from pixel coordinates (I,J) to math (U,V) 2060 LET U = I / 50 - 1.5 2070 LET V = J / 50 - 1 2080 LET X = U 2090 LET Y = V 2100 LET N = 0 2110 REM Inner iteration loop starts here 2120 LET R = X * X 2130 LET Q = Y * Y 2140 IF R + Q > 4 OR N >= L THEN GOTO 2190 2150 LET Y = 2 * X * Y + V 2160 LET X = R - Q + U 2170 LET N = N + 1 2180 GOTO 2120 2190 REM Compute the color to plot 2200 IF N < 10 THEN LET C = "black" ELSE LET C = P[ROUND(8 * (N-10) / (L-10))] 2210 PLOT I, J, C 2220 NEXT J 2230 NEXT I 2240 END 3000 REM Subroutine -- Set up Palette 3010 ARRAY P 3020 LET P[0] = "black" 3030 LET P[1] = "magenta" 3040 LET P[2] = "blue" 3050 LET P[3] = "green" 3060 LET P[4] = "cyan" 3070 LET P[5] = "red" 3080 LET P[6] = "orange" 3090 LET P[7] = "yellow" 3090 LET P[8] = "white" 3100 RETURN </lang>
Run BASIC
<lang Runbasic>'Mandelbrot V4 for RunBasic 'Based on LibertyBasic solution 'copy the code and go to runbasic.com 'http://rosettacode.org/wiki/Mandelbrot_set#Liberty_BASIC 'May 2015 (updated 29 Apr 2018) ' 'Note - we only get so much processing time on the server, so the 'graph is computed in three or four pieces ' WindowWidth = 320 'RunBasic max size 800 x 600 WindowHeight = 320 'print zone -2 to 1 (X) 'print zone -1.5 to 1.5 (Y) a = -1.5 'graph -1.5 to -0.75, first "loop" b = -0.75 'adjust for max processor time (y0 for loop below)
'open "Mandelbrot Set" for graphics_nsb_nf as #w not used in RunBasic
graphic #w, WindowWidth, WindowHeight '#w "trapclose [quit]" not used in RunBasic '#w "down" not used in RunBasic
cls '#w flush()
- w cls("black")
render #w
'#w flush()
input "OK, hit enter to continue"; guess cls
[man_calc] '3/screen size 3/800 = 0.00375 ** 3/790 = 0.0037974 '3/screen size (y) 3/600 = .005 ** 3/590 = 0.0050847 '3/215 = .0139 .0068 = 3/440 cc = 3/299 '
for x0 = -2 to 1 step cc
for y0 = a to b step cc
x = 0
y = 0
iteration = 0
maxIteration = 255
while ( ( x *x +y *y) <=4) and ( iteration <maxIteration)
xtemp =x *x -y *y +x0
y =2 *x *y +y0
x = xtemp
iteration = iteration + 1
wend
if iteration <>maxIteration then
c =iteration
else
c =0
end if
call pSet x0, y0, c
'scan why scan? (wait for user input) with RunBasic ?
next
next
'#w flush() 'what is flush? RunBasic uses the render command. render #w
input "OK, hit enter to continue"; guess cls a = a + 0.75 b = b + 0.75 if b > 1.6 then goto[quit] else goto[man_calc]
sub pSet x, y, c
xScreen = 5+(x +2) /3 * 300 'need positive screen number
yScreen = 5+(y +1.5) /3 * 300 'and 5x5 boarder
if c =0 then
col$ ="red"
else
if c mod 2 =1 then col$ ="lightgray" else col$ ="white"
end if
#w "color "; col$
#w "set "; xScreen; " "; yScreen
end sub
[quit] 'cls print print "This is a Mandelbrot Graph output from www.runbasic.com" render #w print "All done, good bye." end </lang>
Sinclair ZX81 BASIC
Requires at least 2k of RAM.
Glacially slow, but does eventually produce a tolerable low-resolution image (screenshot here). You can adjust the constants in lines 30 and 40 to zoom in on a particular area, if you like. <lang zxbasic> 10 FOR I=0 TO 63
20 FOR J=43 TO 0 STEP -1 30 LET X=(I-52)/31 40 LET Y=(J-22)/31 50 LET XA=0 60 LET YA=0 70 LET ITER=0 80 LET XTEMP=XA*XA-YA*YA+X 90 LET YA=2*XA*YA+Y
100 LET XA=XTEMP 110 LET ITER=ITER+1 120 IF XA*XA+YA*YA<=4 AND ITER<200 THEN GOTO 80 130 IF ITER=200 THEN PLOT I, J 140 NEXT J 150 NEXT I</lang>
SmileBASIC
Generates the points at random, gradually building up the image. <lang smilebasic>X = RNDF()*4-2 Y = RNDF()*4-2@N N = N+16 I = X+S*S-T*T T = Y+S*T*2 S = I IF N < #L&&S*S+T*T < 4 GOTO @N GPSET X*50+99, Y*50+99, RGB(99 XOR N,N,N) EXEC.</lang>
Alternative, based on the QBasic and other BASIC samples.
The 3DS screen is 400 x 240 pixels. SmileBASIC doesn't have +=, -=, etc. but there are INC and DEC statements.
<lang basic>OPTION STRICT
VAR XP, YP, X, Y, X0, Y0, X2, Y2
VAR NEXT_X, IT, C
FOR XP = -200 TO 199
FOR YP = -120 TO 119
X = 0: Y = 0
X0 = XP / 100: Y0 = YP / 100
IT = 0
X2 = X * X: Y2 = Y * Y
WHILE X2 + Y2 <= 4 AND IT < 100
NEXT_X = X2 - Y2 + X0
Y = 2 * X * Y + Y0
X = NEXT_X
X2 = X * X: Y2 = Y * Y
INC IT
WEND
IF IT == 100 THEN C = 0 ELSE C = IT
GPSET XP + 200, YP + 120, RGB((C * 3) MOD 200 + 50, FLOOR(C * 1.2) + 20, C)
NEXT
NEXT</lang>
TI-Basic Color
<lang TI-BASIC> ClrDraw ~2->Xmin:1->Xmax:~1->Ymin:1->Ymax AxesOff FnOff For(A,~2,1,.034 For(B,~1,1,.036 A+B[i]->C DelVar Z9->N While abs(Z)<=2 and N<24 Z^^2+C->Z N+1->N End Pt-On(real(C),imag(C),N End End </lang>
Visual BASIC for Applications on Excel
Based on the BBC BASIC version. Create a spreadsheet with -2 to 2 in row 1 and -2 to 2 in the A column (in steps of your choosing). In the cell B2, call the function with =mandel(B$1,$A2) and copy the cell to all others in the range. Conditionally format the cells to make the colours pleasing (eg based on values, 3-color scale, min value 2 [colour red], midpoint number 10 [green] and highest value black. Then format the cells with the custom type "";"";"" to remove the numbers. <lang VBA> Function mandel(xi As Double, yi As Double)
maxiter = 256 x = 0 y = 0
For i = 1 To maxiter
If ((x * x) + (y * y)) > 4 Then Exit For xt = xi + ((x * x) - (y * y)) y = yi + (2 * x * y) x = xt Next
mandel = i End Function </lang> File:Vbamandel.png Edit: I don't seem to be able to upload the screenshot, so I've shared it here: https://goo.gl/photos/LkezpuQziJPAtdnd9
Yabasic
<lang Yabasic>open window 640, 320 wid = 4 xcenter = -1: ycenter = 0 ms = 0 for xcoord = 0 to 639
for ycoord = 0 to 160
ms = 0
ca =(xcoord-320)/640*wid+xcenter
cb =(ycoord-160)/640*wid+ycenter
x = 0: y=0
for t = 1 to 20
xnew = x*x-y*y+ca
ynew = 2*x*y+cb
x=xnew:y=ynew
magnitudesquared=x*x+y*y
ms = magnitudesquared
if (magnitudesquared > 100) break
//if(magnitudesquared < 100) then : color 0,0,0 : dot xcoord, ycoord : end if
next t
ms = ms+1
if(ms > 250) then
color 32,64,mod(ms,255)
dot xcoord, ycoord
dot xcoord, 320- ycoord
elseif (ms > 150) then
color mod(ms,255),64,32
dot xcoord, ycoord
dot xcoord, 320-ycoord
else
color 0,0,0
dot xcoord, ycoord
dot xcoord, 320-ycoord
end if
next ycoord
next xcoord </lang>
Befunge
Using 14-bit fixed point arithmetic for simplicity and portability. It should work in most interpreters, but the exact output is implementation dependent, and some will be unbearably slow.
X scale is (-2.0, 0.5); Y scale is (-1, 1); Max iterations 94 with the ASCII character set as the "palette".
<lang Befunge>0>:00p58*`#@_0>:01p78vv$$< @^+1g00,+55_v# !`\+*9<>4v$ @v30p20"?~^"< ^+1g10,+*8<$ @>p0\>\::*::882**02g*0v >^ `*:*" d":+*:-*"[Z"+g3 < |< v-*"[Z"+g30*g20**288\--\<# >2**5#>8*:*/00g"P"*58*:*v^ v*288 p20/**288:+*"[Z"+-<: >*%03 p58*:*/01g"3"* v>::^
\_^#!:-1\+-*2*:*85<^
</lang>
- Output:
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Brace
This is a simple Mandelbrot plotter. A longer version based on this smooths colors, and avoids calculating the time-consuming black pixels: http://sam.ai.ki/brace/examples/mandelbrot.d/1 <lang brace>#!/usr/bin/env bx use b
Main(): num outside = 16, ox = -0.5, oy = 0, r = 1.5 long i, max_i = 100, rb_i = 30 space() uint32_t *px = pixel() num d = 2*r/h, x0 = ox-d*w_2, y0 = oy+d*h_2 for(y, 0, h): cmplx c = x0 + (y0-d*y)*I repeat(w): cmplx w = 0 for i=0; i < max_i && cabs(w) < outside; ++i w = w*w + c *px++ = i < max_i ? rainbow(i*359 / rb_i % 360) : black c += d</lang>
An example plot from the longer version:
Brainf***
<lang Brainf***>
A mandelbrot set fractal viewer in brainf*ck written by Erik Bosman
+++++++++++++[->++>>>+++++>++>+<<<<<<]>>>>>++++++>--->>>>>>>>>>+++++++++++++++[[ >>>>>>>>>]+[<<<<<<<<<]>>>>>>>>>-]+[>>>>>>>>[-]>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>[-]+ <<<<<<<+++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>>>>+>>>>>>>>>>>>>>>>>>>>>>>>>> >+<<<<<<<<<<<<<<<<<[<<<<<<<<<]>>>[-]+[>>>>>>[>>>>>>>[-]>>]<<<<<<<<<[<<<<<<<<<]>> >>>>>[-]+<<<<<<++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>>>+<<<<<<+++++++[-[->>> >>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>>>+<<<<<<<<<<<<<<<<[<<<<<<<<<]>>>[[-]>>>>>>[>>>>> >>[-<<<<<<+>>>>>>]<<<<<<[->>>>>>+<<+<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>> [>>>>>>>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+<<+<<<+<<]>>>>>>>>]<<<<<<<<<[<<<<<<< <<]>>>>>>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+<<+<<<<<]>>>>>>>>>+++++++++++++++[[ >>>>>>>>>]+>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+[ >+>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>[-<<<<+>>>>]<<<<[->>>>+<<<<<[->>[ -<<+>>]<<[->>+>>+<<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>>>>>]<<<<<<< <<[>[->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<]>[->>>>>>>>>+<<<<<<<<<]<+>>>>>>>>]<<<<<<<<< [>[-]<->>>>[-<<<<+>[<->-<<<<<<+>>>>>>]<[->+<]>>>>]<<<[->>>+<<<]<+<<<<<<<<<]>>>>> >>>>[>+>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>>[-<<<<<+>>>>>]<<<<<[->>>>>+ <<<<<<[->>>[-<<<+>>>]<<<[->>>+>+<<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>> >>>>>>>]<<<<<<<<<[>>[->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<<]>>[->>>>>>>>>+<<<<<<<<<]<< +>>>>>>>>]<<<<<<<<<[>[-]<->>>>[-<<<<+>[<->-<<<<<<+>>>>>>]<[->+<]>>>>]<<<[->>>+<< <]<+<<<<<<<<<]>>>>>>>>>[>>>>[-<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<+>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>]>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>+++++++++++++++[[>>>> >>>>>]<<<<<<<<<-<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+>>>>>>>>>>>>>>>>>>>>>+<<<[<<<<<< <<<]>>>>>>>>>[>>>[-<<<->>>]+<<<[->>>->[-<<<<+>>>>]<<<<[->>>>+<<<<<<<<<<<<<[<<<<< <<<<]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>>>[-<<<<->>>>]+<<<<[->>>>-<[-<<<+>>>]<<<[-> >>+<<<<<<<<<<<<[<<<<<<<<<]>>>[-]+>>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>>>>>>>>>]<<<<<< <<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]<<<<<<<[->+>>>-<<<<]>>>>>>>>>+++++++++++++++++++ +++++++>>[-<<<<+>>>>]<<<<[->>>>+<<[-]<<]>>[<<<<<<<+<[-<+>>>>+<<[-]]>[-<<[->+>>>- <<<<]>>>]>>>>>>>>>>>>>[>>[-]>[-]>[-]>>>>>]<<<<<<<<<[<<<<<<<<<]>>>[-]>>>>>>[>>>>> [-<<<<+>>>>]<<<<[->>>>+<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>[-<<<<<<<< <+>>>>>>>>>]>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>+++++++++++++++[[>>>>>>>>>]+>[- ]>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+[>+>>>>>>>>]<<< <<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>>[-<<<<<+>>>>>]<<<<<[->>>>>+<<<<<<[->>[-<<+>>]< <[->>+>+<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>>>>>]<<<<<<<<<[>[->>>> >>>>>+<<<<<<<<<]<<<<<<<<<<]>[->>>>>>>>>+<<<<<<<<<]<+>>>>>>>>]<<<<<<<<<[>[-]<->>> [-<<<+>[<->-<<<<<<<+>>>>>>>]<[->+<]>>>]<<[->>+<<]<+<<<<<<<<<]>>>>>>>>>[>>>>>>[-< <<<<+>>>>>]<<<<<[->>>>>+<<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>+>>>>>>>> ]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>>[-<<<<<+>>>>>]<<<<<[->>>>>+<<<<<<[->>[-<<+ >>]<<[->>+>>+<<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>>>>>]<<<<<<<<<[> [->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<]>[->>>>>>>>>+<<<<<<<<<]<+>>>>>>>>]<<<<<<<<<[>[- ]<->>>>[-<<<<+>[<->-<<<<<<+>>>>>>]<[->+<]>>>>]<<<[->>>+<<<]<+<<<<<<<<<]>>>>>>>>> [>>>>[-<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ]>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>[-<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<+> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>]>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>++++++++ +++++++[[>>>>>>>>>]<<<<<<<<<-<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+[>>>>>>>>[-<<<<<<<+ >>>>>>>]<<<<<<<[->>>>>>>+<<<<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>>>>[ -]>>>]<<<<<<<<<[<<<<<<<<<]>>>>+>[-<-<<<<+>>>>>]>[-<<<<<<[->>>>>+<++<<<<]>>>>>[-< <<<<+>>>>>]<->+>]<[->+<]<<<<<[->>>>>+<<<<<]>>>>>>[-]<<<<<<+>>>>[-<<<<->>>>]+<<<< [->>>>->>>>>[>>[-<<->>]+<<[->>->[-<<<+>>>]<<<[->>>+<<<<<<<<<<<<[<<<<<<<<<]>>>[-] +>>>>>>[>>>>>>>>>]>+<]]+>>>[-<<<->>>]+<<<[->>>-<[-<<+>>]<<[->>+<<<<<<<<<<<[<<<<< <<<<]>>>>[-]+>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>>>>>>>>>]<<<<<<<<]>>>>>>>>]<<<<<<<<< [<<<<<<<<<]>>>>[-<<<<+>>>>]<<<<[->>>>+>>>>>[>+>>[-<<->>]<<[->>+<<]>>>>>>>>]<<<<< <<<+<[>[->>>>>+<<<<[->>>>-<<<<<<<<<<<<<<+>>>>>>>>>>>[->>>+<<<]<]>[->>>-<<<<<<<<< <<<<<+>>>>>>>>>>>]<<]>[->>>>+<<<[->>>-<<<<<<<<<<<<<<+>>>>>>>>>>>]<]>[->>>+<<<]<< <<<<<<<<<<]>>>>[-]<<<<]>>>[-<<<+>>>]<<<[->>>+>>>>>>[>+>[-<->]<[->+<]>>>>>>>>]<<< <<<<<+<[>[->>>>>+<<<[->>>-<<<<<<<<<<<<<<+>>>>>>>>>>[->>>>+<<<<]>]<[->>>>-<<<<<<< <<<<<<<+>>>>>>>>>>]<]>>[->>>+<<<<[->>>>-<<<<<<<<<<<<<<+>>>>>>>>>>]>]<[->>>>+<<<< ]<<<<<<<<<<<]>>>>>>+<<<<<<]]>>>>[-<<<<+>>>>]<<<<[->>>>+>>>>>[>>>>>>>>>]<<<<<<<<< [>[->>>>>+<<<<[->>>>-<<<<<<<<<<<<<<+>>>>>>>>>>>[->>>+<<<]<]>[->>>-<<<<<<<<<<<<<< +>>>>>>>>>>>]<<]>[->>>>+<<<[->>>-<<<<<<<<<<<<<<+>>>>>>>>>>>]<]>[->>>+<<<]<<<<<<< <<<<<]]>[-]>>[-]>[-]>>>>>[>>[-]>[-]>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>>>[-< <<<+>>>>]<<<<[->>>>+<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>+++++++++++++++[ [>>>>>>>>>]+>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+ [>+>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>[-<<<<+>>>>]<<<<[->>>>+<<<<<[->> [-<<+>>]<<[->>+>+<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>>>>>]<<<<<<<< <[>[->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<]>[->>>>>>>>>+<<<<<<<<<]<+>>>>>>>>]<<<<<<<<<[ >[-]<->>>[-<<<+>[<->-<<<<<<<+>>>>>>>]<[->+<]>>>]<<[->>+<<]<+<<<<<<<<<]>>>>>>>>>[ >>>[-<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>]> >>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>[-]>>>>+++++++++++++++[[>>>>>>>>>]<<<<<<<<<-<<<<< <<<<[<<<<<<<<<]>>>>>>>>>-]+[>>>[-<<<->>>]+<<<[->>>->[-<<<<+>>>>]<<<<[->>>>+<<<<< <<<<<<<<[<<<<<<<<<]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>>>[-<<<<->>>>]+<<<<[->>>>-<[- <<<+>>>]<<<[->>>+<<<<<<<<<<<<[<<<<<<<<<]>>>[-]+>>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>> >>>>>>>]<<<<<<<<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>[-<<<+>>>]<<<[->>>+>>>>>>[>+>>> [-<<<->>>]<<<[->>>+<<<]>>>>>>>>]<<<<<<<<+<[>[->+>[-<-<<<<<<<<<<+>>>>>>>>>>>>[-<< +>>]<]>[-<<-<<<<<<<<<<+>>>>>>>>>>>>]<<<]>>[-<+>>[-<<-<<<<<<<<<<+>>>>>>>>>>>>]<]> [-<<+>>]<<<<<<<<<<<<<]]>>>>[-<<<<+>>>>]<<<<[->>>>+>>>>>[>+>>[-<<->>]<<[->>+<<]>> >>>>>>]<<<<<<<<+<[>[->+>>[-<<-<<<<<<<<<<+>>>>>>>>>>>[-<+>]>]<[-<-<<<<<<<<<<+>>>> >>>>>>>]<<]>>>[-<<+>[-<-<<<<<<<<<<+>>>>>>>>>>>]>]<[-<+>]<<<<<<<<<<<<]>>>>>+<<<<< ]>>>>>>>>>[>>>[-]>[-]>[-]>>>>]<<<<<<<<<[<<<<<<<<<]>>>[-]>[-]>>>>>[>>>>>>>[-<<<<< <+>>>>>>]<<<<<<[->>>>>>+<<<<+<<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>+>[-<-<<<<+>>>> >]>>[-<<<<<<<[->>>>>+<++<<<<]>>>>>[-<<<<<+>>>>>]<->+>>]<<[->>+<<]<<<<<[->>>>>+<< <<<]+>>>>[-<<<<->>>>]+<<<<[->>>>->>>>>[>>>[-<<<->>>]+<<<[->>>-<[-<<+>>]<<[->>+<< <<<<<<<<<[<<<<<<<<<]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>[-<<->>]+<<[->>->[-<<<+>>>]< <<[->>>+<<<<<<<<<<<<[<<<<<<<<<]>>>[-]+>>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>>>>>>>>>]< <<<<<<<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>[-<<<+>>>]<<<[->>>+>>>>>>[>+>[-<->]<[->+ <]>>>>>>>>]<<<<<<<<+<[>[->>>>+<<[->>-<<<<<<<<<<<<<+>>>>>>>>>>[->>>+<<<]>]<[->>>- <<<<<<<<<<<<<+>>>>>>>>>>]<]>>[->>+<<<[->>>-<<<<<<<<<<<<<+>>>>>>>>>>]>]<[->>>+<<< ]<<<<<<<<<<<]>>>>>[-]>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+<<+<<<<<]]>>>>[-<<<<+> >>>]<<<<[->>>>+>>>>>[>+>>[-<<->>]<<[->>+<<]>>>>>>>>]<<<<<<<<+<[>[->>>>+<<<[->>>- <<<<<<<<<<<<<+>>>>>>>>>>>[->>+<<]<]>[->>-<<<<<<<<<<<<<+>>>>>>>>>>>]<<]>[->>>+<<[ ->>-<<<<<<<<<<<<<+>>>>>>>>>>>]<]>[->>+<<]<<<<<<<<<<<<]]>>>>[-]<<<<]>>>>[-<<<<+>> >>]<<<<[->>>>+>[-]>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+<<+<<<<<]>>>>>>>>>[>>>>>> >>>]<<<<<<<<<[>[->>>>+<<<[->>>-<<<<<<<<<<<<<+>>>>>>>>>>>[->>+<<]<]>[->>-<<<<<<<< <<<<<+>>>>>>>>>>>]<<]>[->>>+<<[->>-<<<<<<<<<<<<<+>>>>>>>>>>>]<]>[->>+<<]<<<<<<<< <<<<]]>>>>>>>>>[>>[-]>[-]>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>[-]>[-]>>>>>[>>>>>[-<<<<+ >>>>]<<<<[->>>>+<<<+<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>>>>[-<<<<<+>>>>> ]<<<<<[->>>>>+<<<+<<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>+++++++++++++++[[>>>> >>>>>]+>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]>[-]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+[>+>> >>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>[-<<<<+>>>>]<<<<[->>>>+<<<<<[->>[-<<+ >>]<<[->>+>>+<<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>>>>>]<<<<<<<<<[> [->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<]>[->>>>>>>>>+<<<<<<<<<]<+>>>>>>>>]<<<<<<<<<[>[- ]<->>>>[-<<<<+>[<->-<<<<<<+>>>>>>]<[->+<]>>>>]<<<[->>>+<<<]<+<<<<<<<<<]>>>>>>>>> [>+>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>->>>>>[-<<<<<+>>>>>]<<<<<[->>>>>+<<<< <<[->>>[-<<<+>>>]<<<[->>>+>+<<<<]+>>>>>>>>>]<<<<<<<<[<<<<<<<<<]]>>>>>>>>>[>>>>>> >>>]<<<<<<<<<[>>[->>>>>>>>>+<<<<<<<<<]<<<<<<<<<<<]>>[->>>>>>>>>+<<<<<<<<<]<<+>>> >>>>>]<<<<<<<<<[>[-]<->>>>[-<<<<+>[<->-<<<<<<+>>>>>>]<[->+<]>>>>]<<<[->>>+<<<]<+ <<<<<<<<<]>>>>>>>>>[>>>>[-<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<+>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>]>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>+++++++++++++++[[>>>>>>>> >]<<<<<<<<<-<<<<<<<<<[<<<<<<<<<]>>>>>>>>>-]+>>>>>>>>>>>>>>>>>>>>>+<<<[<<<<<<<<<] >>>>>>>>>[>>>[-<<<->>>]+<<<[->>>->[-<<<<+>>>>]<<<<[->>>>+<<<<<<<<<<<<<[<<<<<<<<< ]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>>>[-<<<<->>>>]+<<<<[->>>>-<[-<<<+>>>]<<<[->>>+< <<<<<<<<<<<[<<<<<<<<<]>>>[-]+>>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>>>>>>>>>]<<<<<<<<]> >>>>>>>]<<<<<<<<<[<<<<<<<<<]>>->>[-<<<<+>>>>]<<<<[->>>>+<<[-]<<]>>]<<+>>>>[-<<<< ->>>>]+<<<<[->>>>-<<<<<<.>>]>>>>[-<<<<<<<.>>>>>>>]<<<[-]>[-]>[-]>[-]>[-]>[-]>>>[ >[-]>[-]>[-]>[-]>[-]>[-]>>>]<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>>>[-]>>>>]<<<<<<<<< [<<<<<<<<<]>+++++++++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>+>>>>>>>>>+<<<<<<<< <<<<<<[<<<<<<<<<]>>>>>>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+[-]>>[>>>>>>>>>]<<<<< <<<<[>>>>>>>[-<<<<<<+>>>>>>]<<<<<<[->>>>>>+<<<<<<<[<<<<<<<<<]>>>>>>>[-]+>>>]<<<< <<<<<<]]>>>>>>>[-<<<<<<<+>>>>>>>]<<<<<<<[->>>>>>>+>>[>+>>>>[-<<<<->>>>]<<<<[->>> >+<<<<]>>>>>>>>]<<+<<<<<<<[>>>>>[->>+<<]<<<<<<<<<<<<<<]>>>>>>>>>[>>>>>>>>>]<<<<< <<<<[>[-]<->>>>>>>[-<<<<<<<+>[<->-<<<+>>>]<[->+<]>>>>>>>]<<<<<<[->>>>>>+<<<<<<]< +<<<<<<<<<]>>>>>>>-<<<<[-]+<<<]+>>>>>>>[-<<<<<<<->>>>>>>]+<<<<<<<[->>>>>>>->>[>> >>>[->>+<<]>>>>]<<<<<<<<<[>[-]<->>>>>>>[-<<<<<<<+>[<->-<<<+>>>]<[->+<]>>>>>>>]<< <<<<[->>>>>>+<<<<<<]<+<<<<<<<<<]>+++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>+<<< <<[<<<<<<<<<]>>>>>>>>>[>>>>>[-<<<<<->>>>>]+<<<<<[->>>>>->>[-<<<<<<<+>>>>>>>]<<<< <<<[->>>>>>>+<<<<<<<<<<<<<<<<[<<<<<<<<<]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>>>>>>[-< <<<<<<->>>>>>>]+<<<<<<<[->>>>>>>-<<[-<<<<<+>>>>>]<<<<<[->>>>>+<<<<<<<<<<<<<<[<<< <<<<<<]>>>[-]+>>>>>>[>>>>>>>>>]>[-]+<]]+>[-<[>>>>>>>>>]<<<<<<<<]>>>>>>>>]<<<<<<< <<[<<<<<<<<<]>>>>[-]<<<+++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>-<<<<<[<<<<<<< <<]]>>>]<<<<.>>>>>>>>>>[>>>>>>[-]>>>]<<<<<<<<<[<<<<<<<<<]>++++++++++[-[->>>>>>>> >+<<<<<<<<<]>>>>>>>>>]>>>>>+>>>>>>>>>+<<<<<<<<<<<<<<<[<<<<<<<<<]>>>>>>>>[-<<<<<< <<+>>>>>>>>]<<<<<<<<[->>>>>>>>+[-]>[>>>>>>>>>]<<<<<<<<<[>>>>>>>>[-<<<<<<<+>>>>>> >]<<<<<<<[->>>>>>>+<<<<<<<<[<<<<<<<<<]>>>>>>>>[-]+>>]<<<<<<<<<<]]>>>>>>>>[-<<<<< <<<+>>>>>>>>]<<<<<<<<[->>>>>>>>+>[>+>>>>>[-<<<<<->>>>>]<<<<<[->>>>>+<<<<<]>>>>>> >>]<+<<<<<<<<[>>>>>>[->>+<<]<<<<<<<<<<<<<<<]>>>>>>>>>[>>>>>>>>>]<<<<<<<<<[>[-]<- >>>>>>>>[-<<<<<<<<+>[<->-<<+>>]<[->+<]>>>>>>>>]<<<<<<<[->>>>>>>+<<<<<<<]<+<<<<<< <<<]>>>>>>>>-<<<<<[-]+<<<]+>>>>>>>>[-<<<<<<<<->>>>>>>>]+<<<<<<<<[->>>>>>>>->[>>> >>>[->>+<<]>>>]<<<<<<<<<[>[-]<->>>>>>>>[-<<<<<<<<+>[<->-<<+>>]<[->+<]>>>>>>>>]<< <<<<<[->>>>>>>+<<<<<<<]<+<<<<<<<<<]>+++++[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>> +>>>>>>>>>>>>>>>>>>>>>>>>>>>+<<<<<<[<<<<<<<<<]>>>>>>>>>[>>>>>>[-<<<<<<->>>>>>]+< <<<<<[->>>>>>->>[-<<<<<<<<+>>>>>>>>]<<<<<<<<[->>>>>>>>+<<<<<<<<<<<<<<<<<[<<<<<<< <<]>>>>[-]+>>>>>[>>>>>>>>>]>+<]]+>>>>>>>>[-<<<<<<<<->>>>>>>>]+<<<<<<<<[->>>>>>>> -<<[-<<<<<<+>>>>>>]<<<<<<[->>>>>>+<<<<<<<<<<<<<<<[<<<<<<<<<]>>>[-]+>>>>>>[>>>>>> >>>]>[-]+<]]+>[-<[>>>>>>>>>]<<<<<<<<]>>>>>>>>]<<<<<<<<<[<<<<<<<<<]>>>>[-]<<<++++ +[-[->>>>>>>>>+<<<<<<<<<]>>>>>>>>>]>>>>>->>>>>>>>>>>>>>>>>>>>>>>>>>>-<<<<<<[<<<< <<<<<]]>>>] </lang>
- Output:
AAAAAAAAAAAAAAAABBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDEGFFEEEEDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAAAAAABBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDEEEFGIIGFFEEEDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAAAABBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDEEEEFFFI KHGGGHGEDDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAAABBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEEFFGHIMTKLZOGFEEDDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAABBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEEEFGGHHIKPPKIHGFFEEEDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBBBB AAAAAAAAAABBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEEFFGHIJKS X KHHGFEEEEEDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBB AAAAAAAAABBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEEFFGQPUVOTY ZQL[MHFEEEEEEEDDDDDDDCCCCCCCCCCCBBBBBBBBBBBBBB AAAAAAAABBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEFFFFFGGHJLZ UKHGFFEEEEEEEEDDDDDCCCCCCCCCCCCBBBBBBBBBBBB AAAAAAABBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEFFFFFFGGGGHIKP KHHGGFFFFEEEEEEDDDDDCCCCCCCCCCCBBBBBBBBBBB AAAAAAABBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDEEEEEFGGHIIHHHHHIIIJKMR VMKJIHHHGFFFFFFGSGEDDDDCCCCCCCCCCCCBBBBBBBBB AAAAAABBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDEEEEEEFFGHK MKJIJO N R X YUSR PLV LHHHGGHIOJGFEDDDCCCCCCCCCCCCBBBBBBBB AAAAABBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDEEEEEEEEEFFFFGH O TN S NKJKR LLQMNHEEDDDCCCCCCCCCCCCBBBBBBB AAAAABBCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDEEEEEEEEEEEEFFFFFGHHIN Q UMWGEEEDDDCCCCCCCCCCCCBBBBBB AAAABBCCCCCCCCCCCCCCCCCCCCCCCCCDDDDEEEEEEEEEEEEEEEFFFFFFGHIJKLOT [JGFFEEEDDCCCCCCCCCCCCCBBBBB AAAABCCCCCCCCCCCCCCCCCCCCCCDDDDEEEEEEEEEEEEEEEEFFFFFFGGHYV RQU QMJHGGFEEEDDDCCCCCCCCCCCCCBBBB AAABCCCCCCCCCCCCCCCCCDDDDDDDEEFJIHFFFFFFFFFFFFFFGGGGGGHIJN JHHGFEEDDDDCCCCCCCCCCCCCBBB AAABCCCCCCCCCCCDDDDDDDDDDEEEEFFHLKHHGGGGHHMJHGGGGGGHHHIKRR UQ L HFEDDDDCCCCCCCCCCCCCCBB AABCCCCCCCCDDDDDDDDDDDEEEEEEFFFHKQMRKNJIJLVS JJKIIIIIIJLR YNHFEDDDDDCCCCCCCCCCCCCBB AABCCCCCDDDDDDDDDDDDEEEEEEEFFGGHIJKOU O O PR LLJJJKL OIHFFEDDDDDCCCCCCCCCCCCCCB AACCCDDDDDDDDDDDDDEEEEEEEEEFGGGHIJMR RMLMN NTFEEDDDDDDCCCCCCCCCCCCCB AACCDDDDDDDDDDDDEEEEEEEEEFGGGHHKONSZ QPR NJGFEEDDDDDDCCCCCCCCCCCCCC ABCDDDDDDDDDDDEEEEEFFFFFGIPJIIJKMQ VX HFFEEDDDDDDCCCCCCCCCCCCCC ACDDDDDDDDDDEFFFFFFFGGGGHIKZOOPPS HGFEEEDDDDDDCCCCCCCCCCCCCC ADEEEEFFFGHIGGGGGGHHHHIJJLNY TJHGFFEEEDDDDDDDCCCCCCCCCCCCC A PLJHGGFFEEEDDDDDDDCCCCCCCCCCCCC ADEEEEFFFGHIGGGGGGHHHHIJJLNY TJHGFFEEEDDDDDDDCCCCCCCCCCCCC ACDDDDDDDDDDEFFFFFFFGGGGHIKZOOPPS HGFEEEDDDDDDCCCCCCCCCCCCCC ABCDDDDDDDDDDDEEEEEFFFFFGIPJIIJKMQ VX HFFEEDDDDDDCCCCCCCCCCCCCC AACCDDDDDDDDDDDDEEEEEEEEEFGGGHHKONSZ QPR NJGFEEDDDDDDCCCCCCCCCCCCCC AACCCDDDDDDDDDDDDDEEEEEEEEEFGGGHIJMR RMLMN NTFEEDDDDDDCCCCCCCCCCCCCB AABCCCCCDDDDDDDDDDDDEEEEEEEFFGGHIJKOU O O PR LLJJJKL OIHFFEDDDDDCCCCCCCCCCCCCCB AABCCCCCCCCDDDDDDDDDDDEEEEEEFFFHKQMRKNJIJLVS JJKIIIIIIJLR YNHFEDDDDDCCCCCCCCCCCCCBB AAABCCCCCCCCCCCDDDDDDDDDDEEEEFFHLKHHGGGGHHMJHGGGGGGHHHIKRR UQ L HFEDDDDCCCCCCCCCCCCCCBB AAABCCCCCCCCCCCCCCCCCDDDDDDDEEFJIHFFFFFFFFFFFFFFGGGGGGHIJN JHHGFEEDDDDCCCCCCCCCCCCCBBB AAAABCCCCCCCCCCCCCCCCCCCCCCDDDDEEEEEEEEEEEEEEEEFFFFFFGGHYV RQU QMJHGGFEEEDDDCCCCCCCCCCCCCBBBB AAAABBCCCCCCCCCCCCCCCCCCCCCCCCCDDDDEEEEEEEEEEEEEEEFFFFFFGHIJKLOT [JGFFEEEDDCCCCCCCCCCCCCBBBBB AAAAABBCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDEEEEEEEEEEEEFFFFFGHHIN Q UMWGEEEDDDCCCCCCCCCCCCBBBBBB AAAAABBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDEEEEEEEEEFFFFGH O TN S NKJKR LLQMNHEEDDDCCCCCCCCCCCCBBBBBBB AAAAAABBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDEEEEEEFFGHK MKJIJO N R X YUSR PLV LHHHGGHIOJGFEDDDCCCCCCCCCCCCBBBBBBBB AAAAAAABBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDEEEEEFGGHIIHHHHHIIIJKMR VMKJIHHHGFFFFFFGSGEDDDDCCCCCCCCCCCCBBBBBBBBB AAAAAAABBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEFFFFFFGGGGHIKP KHHGGFFFFEEEEEEDDDDDCCCCCCCCCCCBBBBBBBBBBB AAAAAAAABBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEFFFFFGGHJLZ UKHGFFEEEEEEEEDDDDDCCCCCCCCCCCCBBBBBBBBBBBB AAAAAAAAABBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEEFFGQPUVOTY ZQL[MHFEEEEEEEDDDDDDDCCCCCCCCCCCBBBBBBBBBBBBBB AAAAAAAAAABBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDEEEEEEFFGHIJKS X KHHGFEEEEEDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBB AAAAAAAAAAABBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEEEFGGHHIKPPKIHGFFEEEDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBBBB AAAAAAAAAAAABBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDEEEEEFFGHIMTKLZOGFEEDDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAAAABBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDEEEEFFFI KHGGGHGEDDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBB AAAAAAAAAAAAAAABBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDEEEFGIIGFFEEEDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBB
C
PPM non interactive
Here is one file program. It directly creates ppm file. <lang C> /*
c program: -------------------------------- 1. draws Mandelbrot set for Fc(z)=z*z +c using Mandelbrot algorithm ( boolean escape time ) ------------------------------- 2. technique of creating ppm file is based on the code of Claudio Rocchini http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg create 24 bit color graphic file , portable pixmap file = PPM see http://en.wikipedia.org/wiki/Portable_pixmap to see the file use external application ( graphic viewer) */ #include <stdio.h> #include <math.h> int main() { /* screen ( integer) coordinate */ int iX,iY; const int iXmax = 800; const int iYmax = 800; /* world ( double) coordinate = parameter plane*/ double Cx,Cy; const double CxMin=-2.5; const double CxMax=1.5; const double CyMin=-2.0; const double CyMax=2.0; /* */ double PixelWidth=(CxMax-CxMin)/iXmax; double PixelHeight=(CyMax-CyMin)/iYmax; /* color component ( R or G or B) is coded from 0 to 255 */ /* it is 24 bit color RGB file */ const int MaxColorComponentValue=255; FILE * fp; char *filename="new1.ppm"; char *comment="# ";/* comment should start with # */ static unsigned char color[3]; /* Z=Zx+Zy*i ; Z0 = 0 */ double Zx, Zy; double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */ /* */ int Iteration; const int IterationMax=200; /* bail-out value , radius of circle ; */ const double EscapeRadius=2; double ER2=EscapeRadius*EscapeRadius; /*create new file,give it a name and open it in binary mode */ fp= fopen(filename,"wb"); /* b - binary mode */ /*write ASCII header to the file*/ fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue); /* compute and write image data bytes to the file*/ for(iY=0;iY<iYmax;iY++) { Cy=CyMin + iY*PixelHeight; if (fabs(Cy)< PixelHeight/2) Cy=0.0; /* Main antenna */ for(iX=0;iX<iXmax;iX++) { Cx=CxMin + iX*PixelWidth; /* initial value of orbit = critical point Z= 0 */ Zx=0.0; Zy=0.0; Zx2=Zx*Zx; Zy2=Zy*Zy; /* */ for (Iteration=0;Iteration<IterationMax && ((Zx2+Zy2)<ER2);Iteration++) { Zy=2*Zx*Zy + Cy; Zx=Zx2-Zy2 +Cx; Zx2=Zx*Zx; Zy2=Zy*Zy; }; /* compute pixel color (24 bit = 3 bytes) */ if (Iteration==IterationMax) { /* interior of Mandelbrot set = black */ color[0]=0; color[1]=0; color[2]=0; } else { /* exterior of Mandelbrot set = white */ color[0]=255; /* Red*/ color[1]=255; /* Green */ color[2]=255;/* Blue */ }; /*write color to the file*/ fwrite(color,1,3,fp); } } fclose(fp); return 0; }</lang>
PPM Interactive
Infinitely zoomable OpenGL program. Adjustable colors, max iteration, black and white, screen dump, etc. Compile with gcc mandelbrot.c -lglut -lGLU -lGL -lm
- OpenBSD users, install freeglut package, and compile with
make mandelbrot CPPFLAGS='-I/usr/local/include `pkg-config glu --cflags`' LDLIBS='-L/usr/local/lib -lglut `pkg-config glu --libs` -lm'
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- include <GL/glut.h>
- include <GL/gl.h>
- include <GL/glu.h>
void set_texture();
typedef struct {unsigned char r, g, b;} rgb_t; rgb_t **tex = 0; int gwin; GLuint texture; int width, height; int tex_w, tex_h; double scale = 1./256; double cx = -.6, cy = 0; int color_rotate = 0; int saturation = 1; int invert = 0; int max_iter = 256;
void render() { double x = (double)width /tex_w, y = (double)height/tex_h;
glClear(GL_COLOR_BUFFER_BIT); glTexEnvi(GL_TEXTURE_ENV, GL_TEXTURE_ENV_MODE, GL_REPLACE);
glBindTexture(GL_TEXTURE_2D, texture);
glBegin(GL_QUADS);
glTexCoord2f(0, 0); glVertex2i(0, 0); glTexCoord2f(x, 0); glVertex2i(width, 0); glTexCoord2f(x, y); glVertex2i(width, height); glTexCoord2f(0, y); glVertex2i(0, height);
glEnd();
glFlush(); glFinish(); }
int dump = 1; void screen_dump() { char fn[100]; int i; sprintf(fn, "screen%03d.ppm", dump++); FILE *fp = fopen(fn, "w"); fprintf(fp, "P6\n%d %d\n255\n", width, height); for (i = height - 1; i >= 0; i--) fwrite(tex[i], 1, width * 3, fp); fclose(fp); printf("%s written\n", fn); }
void keypress(unsigned char key, int x, int y) { switch(key) { case 'q': glFinish(); glutDestroyWindow(gwin); return; case 27: scale = 1./256; cx = -.6; cy = 0; break;
case 'r': color_rotate = (color_rotate + 1) % 6; break;
case '>': case '.': max_iter += 128; if (max_iter > 1 << 15) max_iter = 1 << 15; printf("max iter: %d\n", max_iter); break;
case '<': case ',': max_iter -= 128; if (max_iter < 128) max_iter = 128; printf("max iter: %d\n", max_iter); break;
case 'c': saturation = 1 - saturation; break;
case 's': screen_dump(); return; case 'z': max_iter = 4096; break; case 'x': max_iter = 128; break; case ' ': invert = !invert; } set_texture(); }
void hsv_to_rgb(int hue, int min, int max, rgb_t *p) { if (min == max) max = min + 1; if (invert) hue = max - (hue - min); if (!saturation) { p->r = p->g = p->b = 255 * (max - hue) / (max - min); return; } double h = fmod(color_rotate + 1e-4 + 4.0 * (hue - min) / (max - min), 6);
- define VAL 255
double c = VAL * saturation; double X = c * (1 - fabs(fmod(h, 2) - 1));
p->r = p->g = p->b = 0;
switch((int)h) { case 0: p->r = c; p->g = X; return; case 1: p->r = X; p->g = c; return; case 2: p->g = c; p->b = X; return; case 3: p->g = X; p->b = c; return; case 4: p->r = X; p->b = c; return; default:p->r = c; p->b = X; } }
void calc_mandel() { int i, j, iter, min, max; rgb_t *px; double x, y, zx, zy, zx2, zy2; min = max_iter; max = 0; for (i = 0; i < height; i++) { px = tex[i]; y = (i - height/2) * scale + cy; for (j = 0; j < width; j++, px++) { x = (j - width/2) * scale + cx; iter = 0;
zx = hypot(x - .25, y); if (x < zx - 2 * zx * zx + .25) iter = max_iter; if ((x + 1)*(x + 1) + y * y < 1/16) iter = max_iter;
zx = zy = zx2 = zy2 = 0; for (; iter < max_iter && zx2 + zy2 < 4; iter++) { zy = 2 * zx * zy + y; zx = zx2 - zy2 + x; zx2 = zx * zx; zy2 = zy * zy; } if (iter < min) min = iter; if (iter > max) max = iter; *(unsigned short *)px = iter; } }
for (i = 0; i < height; i++) for (j = 0, px = tex[i]; j < width; j++, px++) hsv_to_rgb(*(unsigned short*)px, min, max, px); }
void alloc_tex() { int i, ow = tex_w, oh = tex_h;
for (tex_w = 1; tex_w < width; tex_w <<= 1); for (tex_h = 1; tex_h < height; tex_h <<= 1);
if (tex_h != oh || tex_w != ow) tex = realloc(tex, tex_h * tex_w * 3 + tex_h * sizeof(rgb_t*));
for (tex[0] = (rgb_t *)(tex + tex_h), i = 1; i < tex_h; i++) tex[i] = tex[i - 1] + tex_w; }
void set_texture() { alloc_tex(); calc_mandel();
glEnable(GL_TEXTURE_2D); glBindTexture(GL_TEXTURE_2D, texture); glTexImage2D(GL_TEXTURE_2D, 0, 3, tex_w, tex_h, 0, GL_RGB, GL_UNSIGNED_BYTE, tex[0]);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MIN_FILTER, GL_NEAREST); glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, GL_NEAREST); render(); }
void mouseclick(int button, int state, int x, int y) { if (state != GLUT_UP) return;
cx += (x - width / 2) * scale; cy -= (y - height/ 2) * scale;
switch(button) { case GLUT_LEFT_BUTTON: /* zoom in */ if (scale > fabs(x) * 1e-16 && scale > fabs(y) * 1e-16) scale /= 2; break; case GLUT_RIGHT_BUTTON: /* zoom out */ scale *= 2; break; /* any other button recenters */ } set_texture(); }
void resize(int w, int h)
{
printf("resize %d %d\n", w, h);
width = w;
height = h;
glViewport(0, 0, w, h); glOrtho(0, w, 0, h, -1, 1);
set_texture(); }
void init_gfx(int *c, char **v) { glutInit(c, v); glutInitDisplayMode(GLUT_RGB); glutInitWindowSize(640, 480);
gwin = glutCreateWindow("Mandelbrot"); glutDisplayFunc(render);
glutKeyboardFunc(keypress); glutMouseFunc(mouseclick); glutReshapeFunc(resize); glGenTextures(1, &texture); set_texture(); }
int main(int c, char **v) { init_gfx(&c, v); printf("keys:\n\tr: color rotation\n\tc: monochrome\n\ts: screen dump\n\t" "<, >: decrease/increase max iteration\n\tq: quit\n\tmouse buttons to zoom\n");
glutMainLoop(); return 0; }</lang>
ASCII
Not mine, found it on Ken Perlin's homepage, this deserves a place here to illustrate how awesome C can be: <lang C> main(k){float i,j,r,x,y=-16;while(puts(""),y++<15)for(x =0;x++<84;putchar(" .:-;!/>)|&IH%*#"[k&15]))for(i=k=r=0; j=r*r-i*i-2+x/25,i=2*r*i+y/10,j*j+i*i<11&&k++<111;r=j);} </lang> There may be warnings on compiling but disregard them, the output will be produced nevertheless. Such programs are called obfuscated and C excels when it comes to writing such cryptic programs. Google IOCCC for more.
.............::::::::::::::::::::::::::::::::::::::::::::::::.......................
.........::::::::::::::::::::::::::::::::::::::::::::::::::::::::...................
.....::::::::::::::::::::::::::::::::::-----------:::::::::::::::::::...............
...:::::::::::::::::::::::::::::------------------------:::::::::::::::.............
:::::::::::::::::::::::::::-------------;;;!:H!!;;;--------:::::::::::::::..........
::::::::::::::::::::::::-------------;;;;!!/>&*|I !;;;--------::::::::::::::........
::::::::::::::::::::-------------;;;;;;!!/>)|.*#|>/!!;;;;-------::::::::::::::......
::::::::::::::::-------------;;;;;;!!!!//>|: !:|//!!!;;;;-----::::::::::::::.....
::::::::::::------------;;;;;;;!!/>)I>>)||I# H&))>////*!;;-----:::::::::::::....
::::::::----------;;;;;;;;;;!!!//)H: #| IH&*I#/;;-----:::::::::::::...
:::::---------;;;;!!!!!!!!!!!//>|.H: #I>/!;;-----:::::::::::::..
:----------;;;;!/||>//>>>>//>>)|% %|&/!;;----::::::::::::::.
--------;;;;;!!//)& .;I*-H#&||&/ *)/!;;-----::::::::::::::
-----;;;;;!!!//>)IH:- ## #&!!;;-----::::::::::::::
;;;;!!!!!///>)H%.** * )/!;;;------:::::::::::::
&)/!!;;;------:::::::::::::
;;;;!!!!!///>)H%.** * )/!;;;------:::::::::::::
-----;;;;;!!!//>)IH:- ## #&!!;;-----::::::::::::::
--------;;;;;!!//)& .;I*-H#&||&/ *)/!;;-----::::::::::::::
:----------;;;;!/||>//>>>>//>>)|% %|&/!;;----::::::::::::::.
:::::---------;;;;!!!!!!!!!!!//>|.H: #I>/!;;-----:::::::::::::..
::::::::----------;;;;;;;;;;!!!//)H: #| IH&*I#/;;-----:::::::::::::...
::::::::::::------------;;;;;;;!!/>)I>>)||I# H&))>////*!;;-----:::::::::::::....
::::::::::::::::-------------;;;;;;!!!!//>|: !:|//!!!;;;;-----::::::::::::::.....
::::::::::::::::::::-------------;;;;;;!!/>)|.*#|>/!!;;;;-------::::::::::::::......
::::::::::::::::::::::::-------------;;;;!!/>&*|I !;;;--------::::::::::::::........
:::::::::::::::::::::::::::-------------;;;!:H!!;;;--------:::::::::::::::..........
...:::::::::::::::::::::::::::::------------------------:::::::::::::::.............
.....::::::::::::::::::::::::::::::::::-----------:::::::::::::::::::...............
.........::::::::::::::::::::::::::::::::::::::::::::::::::::::::...................
.............::::::::::::::::::::::::::::::::::::::::::::::::.......................
C#
<lang csharp>using System; using System.Drawing; using System.Drawing.Imaging; using System.Threading; using System.Windows.Forms;
/// <summary> /// Generates bitmap of Mandelbrot Set and display it on the form. /// </summary> public class MandelbrotSetForm : Form {
const double MaxValueExtent = 2.0; Thread thread;
static double CalcMandelbrotSetColor(ComplexNumber c)
{
// from http://en.wikipedia.org/w/index.php?title=Mandelbrot_set
const int MaxIterations = 1000;
const double MaxNorm = MaxValueExtent * MaxValueExtent;
int iteration = 0;
ComplexNumber z = new ComplexNumber();
do
{
z = z * z + c;
iteration++;
} while (z.Norm() < MaxNorm && iteration < MaxIterations);
if (iteration < MaxIterations)
return (double)iteration / MaxIterations;
else
return 0; // black
}
static void GenerateBitmap(Bitmap bitmap)
{
double scale = 2 * MaxValueExtent / Math.Min(bitmap.Width, bitmap.Height);
for (int i = 0; i < bitmap.Height; i++)
{
double y = (bitmap.Height / 2 - i) * scale;
for (int j = 0; j < bitmap.Width; j++)
{
double x = (j - bitmap.Width / 2) * scale;
double color = CalcMandelbrotSetColor(new ComplexNumber(x, y));
bitmap.SetPixel(j, i, GetColor(color));
}
}
}
static Color GetColor(double value)
{
const double MaxColor = 256;
const double ContrastValue = 0.2;
return Color.FromArgb(0, 0,
(int)(MaxColor * Math.Pow(value, ContrastValue)));
}
public MandelbrotSetForm()
{
// form creation
this.Text = "Mandelbrot Set Drawing";
this.BackColor = System.Drawing.Color.Black;
this.BackgroundImageLayout = System.Windows.Forms.ImageLayout.Stretch;
this.MaximizeBox = false;
this.StartPosition = FormStartPosition.CenterScreen;
this.FormBorderStyle = FormBorderStyle.FixedDialog;
this.ClientSize = new Size(640, 640);
this.Load += new System.EventHandler(this.MainForm_Load);
}
void MainForm_Load(object sender, EventArgs e)
{
thread = new Thread(thread_Proc);
thread.IsBackground = true;
thread.Start(this.ClientSize);
}
void thread_Proc(object args)
{
// start from small image to provide instant display for user
Size size = (Size)args;
int width = 16;
while (width * 2 < size.Width)
{
int height = width * size.Height / size.Width;
Bitmap bitmap = new Bitmap(width, height, PixelFormat.Format24bppRgb);
GenerateBitmap(bitmap);
this.BeginInvoke(new SetNewBitmapDelegate(SetNewBitmap), bitmap);
width *= 2;
Thread.Sleep(200);
}
// then generate final image
Bitmap finalBitmap = new Bitmap(size.Width, size.Height, PixelFormat.Format24bppRgb);
GenerateBitmap(finalBitmap);
this.BeginInvoke(new SetNewBitmapDelegate(SetNewBitmap), finalBitmap);
}
void SetNewBitmap(Bitmap image)
{
if (this.BackgroundImage != null)
this.BackgroundImage.Dispose();
this.BackgroundImage = image;
}
delegate void SetNewBitmapDelegate(Bitmap image);
static void Main()
{
Application.Run(new MandelbrotSetForm());
}
}
struct ComplexNumber {
public double Re; public double Im;
public ComplexNumber(double re, double im)
{
this.Re = re;
this.Im = im;
}
public static ComplexNumber operator +(ComplexNumber x, ComplexNumber y)
{
return new ComplexNumber(x.Re + y.Re, x.Im + y.Im);
}
public static ComplexNumber operator *(ComplexNumber x, ComplexNumber y)
{
return new ComplexNumber(x.Re * y.Re - x.Im * y.Im,
x.Re * y.Im + x.Im * y.Re);
}
public double Norm()
{
return Re * Re + Im * Im;
}
}</lang>
C++
This generic function assumes that the image can be accessed like a two-dimensional array of colors. It may be passed a true array (in which case the Mandelbrot set will simply be drawn into that array, which then might be saved as image file), or a class which maps the subscript operator to the pixel drawing routine of some graphics library. In the latter case, there must be functions get_first_dimension and get_second_dimension defined for that type, to be found by argument dependent lookup. The code provides those functions for built-in arrays. <lang cpp>#include <cstdlib>
- include <complex>
// get dimensions for arrays template<typename ElementType, std::size_t dim1, std::size_t dim2>
std::size_t get_first_dimension(ElementType (&a)[dim1][dim2])
{
return dim1;
}
template<typename ElementType, std::size_t dim1, std::size_t dim2>
std::size_t get_second_dimension(ElementType (&a)[dim1][dim2])
{
return dim2;
}
template<typename ColorType, typename ImageType>
void draw_Mandelbrot(ImageType& image, //where to draw the image
ColorType set_color, ColorType non_set_color, //which colors to use for set/non-set points
double cxmin, double cxmax, double cymin, double cymax,//the rect to draw in the complex plane
unsigned int max_iterations) //the maximum number of iterations
{
std::size_t const ixsize = get_first_dimension(image);
std::size_t const iysize = get_first_dimension(image);
for (std::size_t ix = 0; ix < ixsize; ++ix)
for (std::size_t iy = 0; iy < iysize; ++iy)
{
std::complex<double> c(cxmin + ix/(ixsize-1.0)*(cxmax-cxmin), cymin + iy/(iysize-1.0)*(cymax-cymin));
std::complex<double> z = 0;
unsigned int iterations;
for (iterations = 0; iterations < max_iterations && std::abs(z) < 2.0; ++iterations)
z = z*z + c;
image[ix][iy] = (iterations == max_iterations) ? set_color : non_set_color;
}
}</lang>
Note this code has not been executed.
A Simple version in CPP. Definitely not as crazy good as the ASCII one in C above.
<lang cpp>#include <stdio.h>
int f(float X, float Y, float x, float y, int n){ return (x*x+y*y<4 && n<100)?1+f(X, Y, x*x-y*y+X, 2*x*y+Y, n+1):0; }
main(){ for(float j=1; j>=-1; j-=.015) for(float i=-2, x; i<=.5; i+=.015, x=f(i, j, 0, 0, 0)) printf("%c%s", x<10?' ':x<20?'.':x<50?':':x<80?'*':'#', i>-2?" ":"\n"); }</lang>
C3
This program produces a BMP as output. <lang c3>module mandelbrot;
extern fn int atoi(char *s); extern fn int printf(char *s, ...); extern fn void putchar(int c);
fn void main(int argc, char **argv) {
int w = atoi(argv[1]); int h = w;
const LIMIT = 2.0; const SQUARE_LIMIT = LIMIT * LIMIT;
printf("P4\n%d %d\n", w, h);
int iter = 50;
int bit_num = 0;
char byte_acc = 0;
for (double y = 0; y < h; y++)
{
for (double x = 0; x < w; x++)
{
double zr;
double zi;
double ti;
double tr;
double cr = (2.0 * x / w - 1.5);
double ci = (2.0 * y / h - 1.0);
for (int i = 0; i < iter && (tr + ti <= SQUARE_LIMIT); i++)
{
zi = 2.0 * zr * zi + ci;
zr = tr - ti + cr;
tr = zr * zr;
ti = zi * zi;
}
byte_acc <<= 1;
if (tr + ti <= SQUARE_LIMIT) byte_acc |= 0x01;
++bit_num;
if (bit_num == 8)
{
putchar(byte_acc);
byte_acc = 0;
bit_num = 0;
}
else if (x == w - 1)
{
byte_acc <<= (8 - w % 8);
putchar(byte_acc);
byte_acc = 0;
bit_num = 0;
}
}
}
}</lang>
Cixl
Displays a zooming Mandelbrot using ANSI graphics.
<lang cixl> use: cx;
define: max 4.0; define: max-iter 570;
let: (max-x max-y) screen-size; let: max-cx $max-x 2.0 /; let: max-cy $max-y 2.0 /; let: rows Stack<Str> new; let: buf Buf new; let: zoom 0 ref;
func: render()()
$rows clear
$max-y 2 / {
let: y;
$buf 0 seek
$max-x {
let: x;
let: (zx zy) 0.0 ref %%;
let: cx $x $max-cx - $zoom deref /;
let: cy $y $max-cy - $zoom deref /;
let: i #max-iter ref;
{
let: nzx $zx deref ** $zy deref ** - $cx +;
$zy $zx deref *2 $zy deref * $cy + set $zx $nzx set
$i &-- set-call
$nzx ** $zy deref ** + #max < $i deref and
} while
let: c $i deref % -7 bsh bor 256 mod;
$c {$x 256 mod $y 256 mod} {0 0} if-else $c new-rgb $buf set-bg
@@s $buf print
} for
$rows $buf str push } for
1 1 #out move-to
$rows {#out print} for
$rows riter {#out print} for;
- out hide-cursor
raw-mode
let: poll Poll new; let: is-done #f ref;
$poll #in {
#in read-char _ $is-done #t set
} on-read
{
$zoom &++ set-call render $poll 0 wait _ $is-done deref !
} while
- out reset-style
- out clear-screen
1 1 #out move-to
- out show-cursor
normal-mode </lang>
Clojure
<lang lisp>(ns mandelbrot
(:refer-clojure :exclude [+ * <])
(:use (clojure.contrib complex-numbers)
(clojure.contrib.generic [arithmetic :only [+ *]]
[comparison :only [<]]
[math-functions :only [abs]])))
(defn mandelbrot? [z]
(loop [c 1
m (iterate #(+ z (* % %)) 0)]
(if (and (> 20 c)
(< (abs (first m)) 2) )
(recur (inc c)
(rest m))
(if (= 20 c) true false))))
(defn mandelbrot []
(for [y (range 1 -1 -0.05)
x (range -2 0.5 0.0315)]
(if (mandelbrot? (complex x y)) "#" " ")))
(println (interpose \newline (map #(apply str %) (partition 80 (mandelbrot))))) </lang>
COBOL
EBCDIC art. <lang cobol>IDENTIFICATION DIVISION. PROGRAM-ID. MANDELBROT-SET-PROGRAM. DATA DIVISION. WORKING-STORAGE SECTION. 01 COMPLEX-ARITHMETIC.
05 X PIC S9V9(9). 05 Y PIC S9V9(9). 05 X-A PIC S9V9(6). 05 X-B PIC S9V9(6). 05 Y-A PIC S9V9(6). 05 X-A-SQUARED PIC S9V9(6). 05 Y-A-SQUARED PIC S9V9(6). 05 SUM-OF-SQUARES PIC S9V9(6). 05 ROOT PIC S9V9(6).
01 LOOP-COUNTERS.
05 I PIC 99. 05 J PIC 99. 05 K PIC 999.
77 PLOT-CHARACTER PIC X. PROCEDURE DIVISION. CONTROL-PARAGRAPH.
PERFORM OUTER-LOOP-PARAGRAPH VARYING I FROM 1 BY 1 UNTIL I IS GREATER THAN 24. STOP RUN.
OUTER-LOOP-PARAGRAPH.
PERFORM INNER-LOOP-PARAGRAPH VARYING J FROM 1 BY 1 UNTIL J IS GREATER THAN 64. DISPLAY .
INNER-LOOP-PARAGRAPH.
MOVE SPACE TO PLOT-CHARACTER. MOVE ZERO TO X-A. MOVE ZERO TO Y-A. MULTIPLY J BY 0.0390625 GIVING X. SUBTRACT 1.5 FROM X. MULTIPLY I BY 0.083333333 GIVING Y. SUBTRACT 1 FROM Y. PERFORM ITERATION-PARAGRAPH VARYING K FROM 1 BY 1 UNTIL K IS GREATER THAN 100 OR PLOT-CHARACTER IS EQUAL TO '#'. DISPLAY PLOT-CHARACTER WITH NO ADVANCING.
ITERATION-PARAGRAPH.
MULTIPLY X-A BY X-A GIVING X-A-SQUARED. MULTIPLY Y-A BY Y-A GIVING Y-A-SQUARED. SUBTRACT Y-A-SQUARED FROM X-A-SQUARED GIVING X-B. ADD X TO X-B. MULTIPLY X-A BY Y-A GIVING Y-A. MULTIPLY Y-A BY 2 GIVING Y-A. SUBTRACT Y FROM Y-A. MOVE X-B TO X-A. ADD X-A-SQUARED TO Y-A-SQUARED GIVING SUM-OF-SQUARES. MOVE FUNCTION SQRT (SUM-OF-SQUARES) TO ROOT. IF ROOT IS GREATER THAN 2 THEN MOVE '#' TO PLOT-CHARACTER.</lang>
- Output:
################################################################
################################# ############################
################################ ###########################
############################## ## ############################
######################## # ######################
######################## ##################
##################### #################
#################### ###############
######## ## ##### ################
####### # ################
###### # #################
####################
###### # #################
####### # ################
######## ## ##### ################
#################### ###############
##################### #################
######################## ##################
######################## # ######################
############################## ## ############################
################################ ###########################
################################# ############################
################################################################
################################################################
Common Lisp
<lang lisp>(defpackage #:mandelbrot
(:use #:cl))
(in-package #:mandelbrot)
(deftype pixel () '(unsigned-byte 8)) (deftype image () '(array pixel))
(defun write-pgm (image filespec)
(declare (image image))
(with-open-file (s filespec :direction :output :element-type 'pixel :if-exists :supersede)
(let* ((width (array-dimension image 1))
(height (array-dimension image 0))
(header (format nil "P5~A~D ~D~A255~A" #\Newline width height #\Newline #\Newline)))
(loop for c across header
do (write-byte (char-code c) s))
(dotimes (row height)
(dotimes (col width)
(write-byte (aref image row col) s))))))
(defparameter *x-max* 800) (defparameter *y-max* 800) (defparameter *cx-min* -2.5) (defparameter *cx-max* 1.5) (defparameter *cy-min* -2.0) (defparameter *cy-max* 2.0) (defparameter *escape-radius* 2) (defparameter *iteration-max* 40)
(defun mandelbrot (filespec)
(let ((pixel-width (/ (- *cx-max* *cx-min*) *x-max*))
(pixel-height (/ (- *cy-max* *cy-min*) *y-max*))
(image (make-array (list *y-max* *x-max*) :element-type 'pixel :initial-element 0)))
(loop for y from 0 below *y-max*
for cy from *cy-min* by pixel-height
do (loop for x from 0 below *x-max*
for cx from *cx-min* by pixel-width
for iteration = (loop with c = (complex cx cy)
for iteration from 0 below *iteration-max*
for z = c then (+ (* z z) c)
while (< (abs z) *escape-radius*)
finally (return iteration))
for pixel = (round (* 255 (/ (- *iteration-max* iteration) *iteration-max*)))
do (setf (aref image y x) pixel)))
(write-pgm image filespec)))</lang>
Cowgol
<lang cowgol>include "cowgol.coh";
const xmin := -8601; const xmax := 2867; const ymin := -4915; const ymax := 4915; const maxiter := 32;
const dx := (xmax-xmin)/79; const dy := (ymax-ymin)/24;
var cy: int16 := ymin; while cy <= ymax loop
var cx: int16 := xmin;
while cx <= xmax loop
var x: int32 := 0;
var y: int32 := 0;
var x2: int32 := 0;
var y2: int32 := 0;
var iter: uint8 := 0;
while iter < maxiter and x2 + y2 <= 16384 loop
y := ((x*y)>>11)+cy as int32;
x := x2-y2+cx as int32;
x2 := (x*x)>>12;
y2 := (y*y)>>12;
iter := iter + 1;
end loop;
print_char(' ' + iter);
cx := cx + dx;
end loop;
print_nl();
cy := cy + dy;
end loop;</lang>
- Output:
!!!!!!!!!!!!!!!"""""""""""""####################################""""""""""""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$$%'+)%%%$$$$$#####"""""""""""
!!!!!!!!!!!"""""""#######################$$$$$$$$%%%&&(+,)++&%$$$$$$######""""""
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*5:/+('&%%$$$$$$#######"""
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''),@@@@@@@,'&%%%%%$$$$########
!!!!!!!"""####################$$$$$$$$%%%&'())((())*,@@@@@@/+))('&&&&)'%$$######
!!!!!!""###################$$$$$%%%%%%&&&'+.@@=/<@@@@@@@@@@@@@@@/++@..93%%$#####
!!!!!"################$$$%%%%%%%%%%&&&&'),+2@@@@@@@@@@@@@@@@@@@@@@@@@1(&&%$$####
!!!!"##########$$$$$%%&(-(''''''''''''(*,5@@@@@@@@@@@@@@@@@@@@@@@@@@@@+)-&%$$###
!!!!####$$$$$$$$%%%%%&'(*-@1.+.@-4+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4-(&%$$$##
!!!!#$$$$$$$$$%%%%%%'''++.6@@@@@@@@@8/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@3(%%$$$$#
!!!#$$$$$$$%&&&&''()/-5.5@@@@@@@@@@@@@>@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@?'&%%$$$$#
!!!(**+/+<523/80/46@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4+)'&&%%$$$$#
!!!#$$$$$$$%&&&&''().-2.@@@@@@@@@@@@@@?@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@'&%%$$$$#
!!!!#$$$$$$$$$%%%%%&'''/,.7@@@@@@@@@;/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@0'%%$$$$#
!!!!####$$$$$$$$%%%%%&'(*-:2.,/?-5+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@4+(&%$$$##
!!!!"##########$$$$$%%&(-(''''(''''''((*,4@@@@@@@@@@@@@@@@@@@@@@@@@@@4+).&%$$###
!!!!!"################$$$%%%%%%%%%%&&&&')<,4@@@@@@@@@@@@@@@@@@@@@@@@@/('&%%$####
!!!!!!""##################$$$$$$%%%%%%&&&'*.@@@0@@@@@@@@@@@@@@@@1,,@//9)%%$#####
!!!!!!!"""####################$$$$$$$$%%%&(())((()**-@@@@@@/+)))'&&&')'%$$######
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''(,@@@@@@@+'&&%%%%%$$$########
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*7@0+('&%%%$$$$$#######"""
!!!!!!!!!!!"""""""######################$$$$$$$$$%%%&&(+-).*&%$$$$$$######""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$%%'3(%%%$$$$$######""""""""""
!!!!!!!!!!!!!!!""""""""""""#####################################""""""""""""""""
D
Textual Version
This uses std.complex because D built-in complex numbers are deprecated.
<lang d>void main() {
import std.stdio, std.complex;
for (real y = -1.2; y < 1.2; y += 0.05) {
for (real x = -2.05; x < 0.55; x += 0.03) {
auto z = 0.complex;
foreach (_; 0 .. 100)
z = z ^^ 2 + complex(x, y);
write(z.abs < 2 ? '#' : '.');
}
writeln;
}
}</lang>
- Output:
....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ................................................................##..................... .............................................................######.................... .............................................................#######................... ..............................................................######................... ..........................................................#.#.###..#.#................. ...................................................##....################.............. ..................................................###.######################.###....... ...................................................############################........ ................................................###############################........ ................................................################################....... .............................................#####################################..... ..............................................###################################...... ..............................##.####.#......####################################...... ..............................###########....####################################...... ............................###############.######################################..... ............................###############.#####################################...... ........................##.#####################################################....... ......#.#####################################################################.......... ........................##.#####################################################....... ............................###############.#####################################...... ............................###############.######################################..... ..............................###########....####################################...... ..............................##.####.#......####################################...... ..............................................###################################...... .............................................#####################################..... ................................................################################....... ................................................###############################........ ...................................................############################........ ..................................................###.######################.###....... ...................................................##....################.............. ..........................................................#.#.###..#.#................. ..............................................................######................... .............................................................#######................... .............................................................######.................... ................................................................##..................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... .......................................................................................
More Functional Textual Version
The output is similar. <lang d>void main() {
import std.stdio, std.complex, std.range, std.algorithm;
foreach (immutable y; iota(-1.2, 1.2, 0.05))
iota(-2.05, 0.55, 0.03).map!(x => 0.complex
.recurrence!((a, n) => a[n - 1] ^^ 2 + complex(x, y))
.drop(100).front.abs < 2 ? '#' : '.').writeln;
}</lang>
Graphical Version
<lang d>import qd;
double lensqr(cdouble c) { return c.re * c.re + c.im * c.im; }
const Limit = 150;
void main() {
screen(640, 480);
for (int y = 0; y < screen.h; ++y) {
flip; events;
for (int x = 0; x < screen.w; ++x) {
auto
c_x = x * 1.0 / screen.w - 0.5,
c_y = y * 1.0 / screen.h - 0.5,
c = c_y * 2.0i + c_x * 3.0 - 1.0,
z = 0.0i + 0.0,
i = 0;
for (; i < Limit; ++i) {
z = z * z + c;
if (lensqr(z) > 4) break;
}
auto value = cast(ubyte) (i * 255.0 / Limit);
pset(x, y, rgb(value, value, value));
}
}
while (true) { flip; events; }
}</lang>
Dart
Implementation in Google Dart works on http://try.dartlang.org/ (as of 10/18/2011) since the language is very new, it may break in the future. The implementation uses a incomplete Complex class supporting operator overloading. <lang>class Complex {
double _r,_i;
Complex(this._r,this._i); double get r => _r; double get i => _i; String toString() => "($r,$i)";
Complex operator +(Complex other) => new Complex(r+other.r,i+other.i);
Complex operator *(Complex other) =>
new Complex(r*other.r-i*other.i,r*other.i+other.r*i);
double abs() => r*r+i*i;
}
void main() {
double start_x=-1.5; double start_y=-1.0; double step_x=0.03; double step_y=0.1;
for(int y=0;y<20;y++) {
String line="";
for(int x=0;x<70;x++) {
Complex c=new Complex(start_x+step_x*x,start_y+step_y*y);
Complex z=new Complex(0.0, 0.0);
for(int i=0;i<100;i++) {
z=z*(z)+c;
if(z.abs()>2) {
break;
}
}
line+=z.abs()>2 ? " " : "*";
}
print(line);
}
}</lang>
Dc
ASCII output
This can be done in a more Dc-ish way, e.g. by moving the loop macros' definitions to the initialisations in the top instead of saving the macro definition of inner loops over and over again in outer loops.
<lang dc> _2.1 sx # xmin = -2.1
0.7 sX # xmax = 0.7
_1.2 sy # ymin = -1.2 1.2 sY # ymax = 1.2
32 sM # maxiter = 32
80 sW # image width 25 sH # image height
8 k # precision
[ q ] sq # quitter helper macro
- for h from 0 to H-1
0 sh [
lh lH =q # quit if H reached
# for w from 0 to W-1 0 sw [ lw lW =q # quit if W reached
# (w,h) -> (R,I) # | | # | ymin + h*(ymax-ymin)/(height-1) # xmin + w*(xmax-xmin)/(width-1)
lX lx - lW 1 - / lw * lx + sR lY ly - lH 1 - / lh * ly + sI
# iterate for (R,I)
0 sr # r:=0 0 si # i:=0 0 sa # a:=0 (r squared) 0 sb # b:=0 (i squared) 0 sm # m:=0
# do while m!=M and a+b=<4
[
lm lM =q # exit if m==M
la lb + 44
2 lr * li * lI + si # i:=2*r*i+I
la lb - lR + sr # r:=a-b+R
lm 1 + sm # m+=1
lr 2 ^ sa # a:=r*r
li 2 ^ sb # b:=i*i
l0 x # loop ] s0 l0 x
lm 32 + P # print "pixel"
lw 1 + sw # w+=1 l1 x # loop ] s1 l1 x
A P # linefeed
lh 1 + sh # h+=1 l2 x # loop
] s2 l2 x</lang>
- Output:
!!!!!!!!!!!!!!!"""""""""""""####################################""""""""""""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$$%'0(%%%$$$$$#####"""""""""""
!!!!!!!!!!!"""""""#######################$$$$$$$$%%%&&(++)++&$$$$$$$######""""""
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*@;/*('&%%$$$$$$#######"""
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''),@@@@@@@+'&%%%%%$$$$########
!!!!!!!"""####################$$$$$$$$%%%&'())((())*-@@@@@@.+))('&&&&+&%$$######
!!!!!!""###################$$$$$%%%%%%&&&'+.@@@08@@@@@@@@@@@@@@@/+,@//@)%%$#####
!!!!!"################$$$%%%%%%%%%%&&&&')-+7@@@@@@@@@@@@@@@@@@@@@@@@@4(&&%$$####
!!!!"##########$$$$$%%&(,('''''''''''((*-5@@@@@@@@@@@@@@@@@@@@@@@@@@@3+)4&%$$###
!!!!####$$$$$$$$%%%%%&'(*-@1.+/@-4+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@3+'&%$$$##
!!!!#$$$$$$$$$%%%%%%'''++.7@@@@@@@@@9/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@<6'%%$$$$#
!!!#$$$$$$$%&&&&''().-2.6@@@@@@@@@@@@@>@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@'&%%$$$$#
!!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@2+)'&&%%$$$$#
!!!#$$$$$$$%&&&&''().-2.6@@@@@@@@@@@@@>@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@'&%%$$$$#
!!!!#$$$$$$$$$%%%%%%'''++.7@@@@@@@@@9/0@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@<6'%%$$$$#
!!!!####$$$$$$$$%%%%%&'(*-@1.+/@-4+))**@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@3+'&%$$$##
!!!!"##########$$$$$%%&(,('''''''''''((*-5@@@@@@@@@@@@@@@@@@@@@@@@@@@3+)4&%$$###
!!!!!"################$$$%%%%%%%%%%&&&&')-+7@@@@@@@@@@@@@@@@@@@@@@@@@4(&&%$$####
!!!!!!""###################$$$$$%%%%%%&&&'+.@@@08@@@@@@@@@@@@@@@/+,@//@)%%$#####
!!!!!!!"""####################$$$$$$$$%%%&'())((())*-@@@@@@.+))('&&&&+&%$$######
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''),@@@@@@@+'&%%%%%$$$$########
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*@;/*('&%%$$$$$$#######"""
!!!!!!!!!!!"""""""#######################$$$$$$$$%%%&&(++)++&$$$$$$$######""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$$%'0(%%%$$$$$#####"""""""""""
!!!!!!!!!!!!!!!"""""""""""""####################################""""""""""""""""
PGM (P5) output
This is a condensed version of the ASCII output variant modified to generate a PGM (P5) image. <lang dc>_2.1 sx 0.7 sX _1.2 sy 1.2 sY 32 sM 640 sW 480 sH 8 k [P5] P A P lW n 32 P lH n A P lM 1 - n A P [ q ] sq 0 sh [
lh lH =q
0 sw
[
lw lW =q
lX lx - lW 1 - / lw * lx + sR
lY ly - lH 1 - / lh * ly + sI
0 sr 0 si 0 sa 0 sb 0 sm
[
lm lM =q
la lb + 4<q
2 lr * li * lI + si
la lb - lR + sr
lm 1 + sm
lr 2 ^ sa
li 2 ^ sb
l0 x
] s0
l0 x
lm 1 - P
lw 1 + sw
l1 x
] s1
l1 x
lh 1 + sh
l2 x
] s2 l2 x</lang>
Delphi
See Pascal.
DWScript
<lang delphi>const maxIter = 256;
var x, y, i : Integer; for y:=-39 to 39 do begin
for x:=-39 to 39 do begin
var c := Complex(y/40-0.5, x/40);
var z := Complex(0, 0);
for i:=1 to maxIter do begin
z := z*z + c;
if Abs(z)>=4 then Break;
end;
if i>=maxIter then
Print('#')
else Print('.');
end;
PrintLn();
end;</lang>
EasyLang
<lang>for y0 range 300
cy = (y0 - 150) / 120
for x0 range 300
cx = (x0 - 220) / 120
x = 0
y = 0
color3 0 0 0
for n range 128
if x * x + y * y > 4
color3 n / 16 0 0
break 1
.
h = x * x - y * y + cx
y = 2 * x * y + cy
x = h
.
move x0 / 3 y0 / 3
rect 0.4 0.4
.
.</lang>
eC
Drawing code: <lang eC>void drawMandelbrot(Bitmap bmp, float range, Complex center, ColorAlpha * palette, int nPalEntries, int nIterations, float scale) {
int x, y;
int w = bmp.width, h = bmp.height;
ColorAlpha * picture = (ColorAlpha *)bmp.picture;
double logOf2 = log(2);
Complex d
{
w > h ? range : range * w / h,
h > w ? range : range * h / w
};
Complex C0 { center.a - d.a/2, center.b - d.b/2 };
Complex C = C0;
double delta = d.a / w;
for(y = 0; y < h; y++, C.a = C0.a, C.b += delta)
{
for(x = 0; x < w; x++, picture++, C.a += delta)
{
Complex Z { };
int i;
double ii = 0;
bool out = false;
double Za2 = Z.a * Z.a, Zb2 = Z.b * Z.b;
for(i = 0; i < nIterations; i++)
{
double z2;
Z = { Za2 - Zb2, 2*Z.a*Z.b };
Z.a += C.a;
Z.b += C.b;
Za2 = Z.a * Z.a, Zb2 = Z.b * Z.b;
z2 = Za2 + Zb2;
if(z2 >= 2*2)
{
ii = (double)(i + 1 - log(0.5 * log(z2)) / logOf2);
out = true;
break;
}
}
if(out)
{
float si = (float)(ii * scale);
int i0 = ((int)si) % nPalEntries;
*picture = palette[i0];
}
else
*picture = black;
}
}
}</lang> Interactive class with Rubberband Zoom: <lang eC>class Mandelbrot : Window {
caption = $"Mandelbrot";
borderStyle = sizable;
hasMaximize = true;
hasMinimize = true;
hasClose = true;
clientSize = { 600, 600 };
Point mouseStart, mouseEnd; bool dragging; bool needUpdate;
float scale;
int nIterations; nIterations = 256;
ColorAlpha * palette;
int nPalEntries;
Complex center { -0.75, 0 };
float range; range = 4;
Bitmap bmp { };
Mandelbrot()
{
static ColorKey keys[] =
{
{ navy, 0.0f },
{ Color { 146, 213, 237 }, 0.198606268f },
{ white, 0.3f },
{ Color { 255, 255, 124 }, 0.444250882f },
{ Color { 255, 100, 0 }, 0.634146333f },
{ navy, 1 }
};
nPalEntries = 30000;
palette = new ColorAlpha[nPalEntries];
scale = nPalEntries / 175.0f;
PaletteGradient(palette, nPalEntries, keys, sizeof(keys)/sizeof(keys[0]), 1.0);
needUpdate = true;
}
~Mandelbrot() { delete palette; }
void OnRedraw(Surface surface)
{
if(needUpdate)
{
drawMandelbrot(bmp, range, center, palette, nPalEntries, nIterations, scale);
needUpdate = false;
}
surface.Blit(bmp, 0,0, 0,0, bmp.width, bmp.height);
if(dragging)
{
surface.foreground = lime;
surface.Rectangle(mouseStart.x, mouseStart.y, mouseEnd.x, mouseEnd.y);
}
}
bool OnLeftButtonDown(int x, int y, Modifiers mods)
{
mouseEnd = mouseStart = { x, y };
Capture();
dragging = true;
Update(null);
return true;
}
bool OnLeftButtonUp(int x, int y, Modifiers mods)
{
if(dragging)
{
int dx = Abs(mouseEnd.x - mouseStart.x), dy = Abs(mouseEnd.y - mouseStart.y);
if(dx > 4 && dy > 4)
{
int w = clientSize.w, h = clientSize.h;
float rangeX = w > h ? range : range * w / h;
float rangeY = h > w ? range : range * h / w;
center.a += ((mouseStart.x + mouseEnd.x) - w) / 2.0f * rangeX / w;
center.b += ((mouseStart.y + mouseEnd.y) - h) / 2.0f * rangeY / h;
range = dy > dx ? dy * range / h : dx * range / w;
needUpdate = true;
Update(null);
}
ReleaseCapture();
dragging = false;
}
return true;
}
bool OnMouseMove(int x, int y, Modifiers mods)
{
if(dragging)
{
mouseEnd = { x, y };
Update(null);
}
return true;
}
bool OnRightButtonDown(int x, int y, Modifiers mods)
{
range = 4;
nIterations = 256;
center = { -0.75, 0 };
needUpdate = true;
Update(null);
return true;
}
void OnResize(int width, int height)
{
bmp.Allocate(null, width, height, 0, pixelFormat888, false);
needUpdate = true;
Update(null);
}
bool OnKeyHit(Key key, unichar ch)
{
switch(key)
{
case space: case keyPadPlus: case plus:
nIterations += 256;
needUpdate = true;
Update(null);
break;
}
return true;
}
}
Mandelbrot mandelbrotForm {};</lang>
EchoLisp
<lang scheme> (lib 'math) ;; fractal function (lib 'plot)
- (fractal z zc n) iterates z
- = z^2 + c, n times
- 100 iterations
(define (mset z) (if (= Infinity (fractal 0 z 100)) Infinity z))
- plot function argument inside square (-2 -2), (2,2)
(plot-z-arg mset -2 -2)
- result here [1]
</lang>
Elixir
<lang elixir>defmodule Mandelbrot do
def set do
xsize = 59
ysize = 21
minIm = -1.0
maxIm = 1.0
minRe = -2.0
maxRe = 1.0
stepX = (maxRe - minRe) / xsize
stepY = (maxIm - minIm) / ysize
Enum.each(0..ysize, fn y ->
im = minIm + stepY * y
Enum.map(0..xsize, fn x ->
re = minRe + stepX * x
62 - loop(0, re, im, re, im, re*re+im*im)
end) |> IO.puts
end)
end
defp loop(n, _, _, _, _, _) when n>=30, do: n
defp loop(n, _, _, _, _, v) when v>4.0, do: n-1
defp loop(n, re, im, zr, zi, _) do
a = zr * zr
b = zi * zi
loop(n+1, re, im, a-b+re, 2*zr*zi+im, a+b)
end
end
Mandelbrot.set</lang>
- Output:
??????=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<=========== ?????===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== ????===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= ???==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== ??==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== ??=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== ?=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== ?<<<<;;;;;:::972456-567763 +9;;<<<<<<<= ?<;;;;;;::::9875& .3 *9;;;<<<<<<= ?;;;;;;::997564' ' 8:;;;<<<<<<= ?::988897735/ &89:;;;<<<<<<= ?::988897735/ &89:;;;<<<<<<= ?;;;;;;::997564' ' 8:;;;<<<<<<= ?<;;;;;;::::9875& .3 *9;;;<<<<<<= ?<<<<;;;;;:::972456-567763 +9;;<<<<<<<= ?=<<<<<<<<;;;:599999999886 %78:;;<<<<<<== ??=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<=== ??==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<==== ???==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<===== ????===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<======= ?????===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======== ??????=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<===========
Emacs Lisp
Text mode
<lang lisp>; === Mandelbrot ============================================
(setq mandel-size (cons 76 34)) (setq xmin -2) (setq xmax .5) (setq ymin -1.2) (setq ymax 1.2) (setq max-iter 20)
(defun mandel-iter-point (x y)
"Run the actual iteration for each point."
(let ((xp 0)
(yp 0)
(it 0)
(xt 0))
(while (and (< (+ (* xp xp) (* yp yp)) 4) (< it max-iter))
(setq xt (+ (* xp xp) (* -1 yp yp) x))
(setq yp (+ (* 2 xp yp) y))
(setq xp xt)
(setq it (1+ it)))
it))
(defun mandel-iter (p)
"Return string for point based on whether inside/outside the set." (let ((it (mandel-iter-point (car p) (cdr p)))) (if (= it max-iter) "*" "-")))
(defun mandel-pos (x y)
"Convert screen coordinates to input coordinates."
(let ((xp (+ xmin (* (- xmax xmin) (/ (float x) (car mandel-size)))))
(yp (+ ymin (* (- ymax ymin) (/ (float y) (cdr mandel-size))))))
(cons xp yp)))
(defun mandel ()
"Plot the Mandelbrot set."
(dotimes (y (cdr mandel-size))
(dotimes (x (car mandel-size))
(if (= x 0)
(insert(format "\n%s" (mandel-iter (mandel-pos x y))))
(insert(format "%s" (mandel-iter (mandel-pos x y))))))))
(mandel)</lang>
- Output:
---------------------------------------------------------------------------- ---------------------------------------------------------------------------- ---------------------------------------------------------------------------- ---------------------------------------------------------*------------------ ---------------------------------------------------------------------------- ----------------------------------------------------**--***--*-------------- -----------------------------------------------------********--------------- ------------------------------------------------------*******--------------- --------------------------------------------*-*--******************--------- --------------------------------------------****************************---- -----------------------------------------*-****************************----- -----------------------------------------*******************************---- ---------------------------------------************************************- --------------------------**--**-*-----**********************************--- -------------------------***********---************************************- ------------------*-----**************************************************-- -------------------*****************************************************---- -*********************************************************************------ -------------------*****************************************************---- ------------------*-----**************************************************-- -------------------------***********---************************************- --------------------------**--**-*-----**********************************--- ---------------------------------------************************************- -----------------------------------------*******************************---- -----------------------------------------*-****************************----- --------------------------------------------****************************---- --------------------------------------------*-*--******************--------- ------------------------------------------------------*******--------------- -----------------------------------------------------********--------------- ----------------------------------------------------**--***--*-------------- ---------------------------------------------------------------------------- ---------------------------------------------------------*------------------ ---------------------------------------------------------------------------- ----------------------------------------------------------------------------
Graphical version
With a few modifications (mandel-size, mandel-iter, string-to-image, mandel-pic), the code above can also render the Mandelbrot fractal to an XPM image and display it directly in the buffer. (You might have to scroll up in Emacs after the function has run to see its output.) <lang lisp>; === Graphical Mandelbrot ============================================
(setq mandel-size (cons 320 300)) (setq xmin -2) (setq xmax .5) (setq ymin -1.2) (setq ymax 1.2) (setq max-iter 20)
(defun mandel-iter-point (x y)
"Run the actual iteration for each point."
(let ((xp 0)
(yp 0)
(it 0)
(xt 0))
(while (and (< (+ (* xp xp) (* yp yp)) 4) (< it max-iter))
(setq xt (+ (* xp xp) (* -1 yp yp) x))
(setq yp (+ (* 2 xp yp) y))
(setq xp xt)
(setq it (1+ it)))
it))
(defun mandel-iter (p)
"Return string for point based on whether inside/outside the set." (let ((it (mandel-iter-point (car p) (cdr p)))) (if (= it max-iter) "*" (if (cl-oddp it) "+" "-"))))
(defun mandel-pos (x y)
"Convert screen coordinates to input coordinates."
(let ((xp (+ xmin (* (- xmax xmin) (/ (float x) (car mandel-size)))))
(yp (+ ymin (* (- ymax ymin) (/ (float y) (cdr mandel-size))))))
(cons xp yp)))
(defun string-to-image (str)
"Convert image data string to XPM image." (create-image (concat (format "/* XPM */
static char * mandel[] = { \"%i %i 3 1\", \"+ c #ff0000\", \"- c #0000ff\", \"* c #000000\"," (car mandel-size) (cdr mandel-size))
str "};") 'xpm t))
(defun mandel-pic ()
"Plot the Mandelbrot set."
(setq all "")
(dotimes (y (cdr mandel-size))
(setq line "")
(dotimes (x (car mandel-size))
(setq line (concat line (mandel-iter (mandel-pos x y)))))
(setq all (concat all "\"" line "\",\n")))
(insert-image (string-to-image all)))
(mandel-pic)</lang>
Erlang
Function seq_float/2 is copied from Andrew Fecheyr's GitHubGist.
Using module complex from Geoff Hulette's GitHub repository
Geoff Hulette's GitHub repository provides two alternative implementations which are very interesting.
<lang erlang> -module(mandelbrot).
-export([test/0]).
magnitude(Z) ->
R = complex:real(Z), I = complex:imaginary(Z), R * R + I * I.
mandelbrot(A, MaxI, Z, I) ->
case (I < MaxI) and (magnitude(Z) < 2.0) of
true ->
NZ = complex:add(complex:mult(Z, Z), A),
mandelbrot(A, MaxI, NZ, I + 1);
false ->
case I of
MaxI ->
$*;
_ ->
$
end
end.
test() ->
lists:map(
fun(S) -> io:format("~s",[S]) end,
[
[
begin
Z = complex:make(X, Y),
mandelbrot(Z, 50, Z, 1)
end
|| X <- seq_float(-2, 0.5, 0.0315)
] ++ "\n"
|| Y <- seq_float(-1,1, 0.05)
] ),
ok.
% ************************************************** % Copied from https://gist.github.com/andruby/241489 % **************************************************
seq_float(Min, Max, Inc, Counter, Acc) when (Counter*Inc + Min) >= Max ->
lists:reverse([Max|Acc]);
seq_float(Min, Max, Inc, Counter, Acc) ->
seq_float(Min, Max, Inc, Counter+1, [Inc * Counter + Min|Acc]).
seq_float(Min, Max, Inc) ->
seq_float(Min, Max, Inc, 0, []).
% ************************************************** </lang>
Output:
**
******
********
******
******** ** *
*** *****************
************************ ***
****************************
******************************
******************************
************************************
* **********************************
** ***** * **********************************
*********** ************************************
************** ************************************
***************************************************
*****************************************************
*****************************************************
*****************************************************
***************************************************
************** ************************************
*********** ************************************
** ***** * **********************************
* **********************************
************************************
******************************
******************************
****************************
************************ ***
*** *****************
******** ** *
******
********
******
**
ERRE
<lang ERRE> PROGRAM MANDELBROT
!$KEY !$INCLUDE="PC.LIB"
BEGIN
SCREEN(7) GR_WINDOW(-2,1.5,2,-1.5) FOR X0=-2 TO 2 STEP 0.01 DO
FOR Y0=-1.5 TO 1.5 STEP 0.01 DO
X=0
Y=0
ITERATION=0
MAX_ITERATION=223
WHILE (X*X+Y*Y<=(2*2) AND ITERATION<MAX_ITERATION) DO
X_TEMP=X*X-Y*Y+X0
Y=2*X*Y+Y0
X=X_TEMP
ITERATION=ITERATION+1
END WHILE
IF ITERATION<>MAX_ITERATION THEN
C=ITERATION
ELSE
C=0
END IF
PSET(X0,Y0,C) END FOR
END FOR END PROGRAM </lang> Note: This is a PC version which uses EGA 16-color 320x200. Graphic commands are taken from PC.LIB library.
F#
<lang fsharp>open System.Drawing open System.Windows.Forms type Complex =
{
re : float;
im : float
}
let cplus (x:Complex) (y:Complex) : Complex =
{
re = x.re + y.re;
im = x.im + y.im
}
let cmult (x:Complex) (y:Complex) : Complex =
{
re = x.re * y.re - x.im * y.im;
im = x.re * y.im + x.im * y.re;
}
let norm (x:Complex) : float =
x.re*x.re + x.im*x.im
type Mandel = class
inherit Form
static member xPixels = 500
static member yPixels = 500
val mutable bmp : Bitmap
member x.mandelbrot xMin xMax yMin yMax maxIter =
let rec mandelbrotIterator z c n =
if (norm z) > 2.0 then false else
match n with
| 0 -> true
| n -> let z' = cplus ( cmult z z ) c in
mandelbrotIterator z' c (n-1)
let dx = (xMax - xMin) / (float (Mandel.xPixels))
let dy = (yMax - yMin) / (float (Mandel.yPixels))
in
for xi = 0 to Mandel.xPixels-1 do
for yi = 0 to Mandel.yPixels-1 do
let c = {re = xMin + (dx * float(xi) ) ;
im = yMin + (dy * float(yi) )} in
if (mandelbrotIterator {re=0.;im=0.;} c maxIter) then
x.bmp.SetPixel(xi,yi,Color.Azure)
else
x.bmp.SetPixel(xi,yi,Color.Black)
done
done
member public x.generate () = x.mandelbrot (-1.5) 0.5 (-1.0) 1.0 200 ; x.Refresh()
new() as x = {bmp = new Bitmap(Mandel.xPixels , Mandel.yPixels)} then
x.Text <- "Mandelbrot set" ;
x.Width <- Mandel.xPixels ;
x.Height <- Mandel.yPixels ;
x.BackgroundImage <- x.bmp;
x.generate();
x.Show();
end
let f = new Mandel() do Application.Run(f)</lang>
Alternate version, applicable to text and GUI
Basic generation code <lang fsharp> let getMandelbrotValues width height maxIter ((xMin,xMax),(yMin,yMax)) =
let mandIter (cr:float,ci:float) =
let next (zr,zi) = (cr + (zr * zr - zi * zi)), (ci + (zr * zi + zi * zr))
let rec loop = function
| step,_ when step=maxIter->0
| step,(zr,zi) when ((zr * zr + zi * zi) > 2.0) -> step
| step,z -> loop ((step + 1), (next z))
loop (0,(0.0, 0.0))
let forPos =
let dx, dy = (xMax - xMin) / (float width), (yMax - yMin) / (float height)
fun y x -> mandIter ((xMin + dx * float(x)), (yMin + dy * float(y)))
[0..height-1] |> List.map(fun y->[0..width-1] |> List.map (forPos y))
</lang>
Text display <lang fsharp> getMandelbrotValues 80 25 50 ((-2.0,1.0),(-1.0,1.0)) |> List.map(fun row-> row |> List.map (function | 0 ->" " |_->".") |> String.concat "") |> List.iter (printfn "%s") </lang>
Results:
- Output:
................................................................................ ................................................................................ ................................................. ............................. ................................................ ........................... ................................................. ........................... ....................................... . ...................... ........................................ ................. .................................... .................. .................................... ................. .......................... ...... ................ ....................... ... ................ ..................... . ................ ................. ................. ................. ................. ..................... . ................ ....................... ... ................ .......................... ...... ................ .................................... ................. .................................... .................. ........................................ ................. ....................................... . ...................... ................................................. ........................... ................................................ ........................... ................................................. ............................. ................................................................................
Graphics display <lang fsharp> open System.Drawing open System.Windows.Forms
let showGraphic (colorForIter: int -> Color) (width: int) (height:int) maxIter view =
new Form()
|> fun frm ->
frm.Width <- width
frm.Height <- height
frm.BackgroundImage <-
new Bitmap(width,height)
|> fun bmp ->
getMandelbrotValues width height maxIter view
|> List.mapi (fun y row->row |> List.mapi (fun x v->((x,y),v))) |> List.collect id
|> List.iter (fun ((x,y),v) -> bmp.SetPixel(x,y,(colorForIter v)))
bmp
frm.Show()
let toColor = (function | 0 -> (0,0,0) | n -> ((31 &&& n) |> fun x->(0, 18 + x * 5, 36 + x * 7))) >> Color.FromArgb
showGraphic toColor 640 480 5000 ((-2.0,1.0),(-1.0,1.0)) </lang>
Factor
<lang Factor> ! with ("::") or without (":") generalizations: ! : [a..b] ( steps a b -- a..b ) 2dup swap - 4 nrot 1 - / <range> ;
- [a..b] ( steps a b -- a..b ) a b b a - steps 1 - / <range> ;
- >char ( n -- c )
dup -1 = [ drop 32 ] [ 26 mod CHAR: a + ] if ;
! iterates z' = z^2 + c, Factor does complex numbers!
- iter ( c z -- z' ) dup * + ;
- unbound ( c -- ? ) absq 4 > ;
- mz ( c max i z -- n )
{
{ [ i max >= ] [ -1 ] }
{ [ z unbound ] [ i ] }
[ c max i 1 + c z iter mz ]
} cond ;
- mandelzahl ( c max -- n ) 0 0 mz ;
- mandel ( w h max -- )
h -1. 1. [a..b] ! range over y
[ w -2. 1. [a..b] ! range over x
[ dupd swap rect> max mandelzahl >char ] map
>string print
drop ! old y
] each
;
70 25 1000 mandel
</lang>
- Output:
bbbbbbbcccccdddddddddddddddddddeeeeeeeffghjpjl feeeeedddddcccccccccccc
bbbbbbccccddddddddddddddddddeeeeeeeefffghikopjhgffeeeeedddddcccccccccc
bbbbbcccddddddddddddddddddeeeeeeeefffggjotx etiigfffeeeeddddddcccccccc
bbbbccddddddddddddddddddeeeeeeeffgggghhjq iihgggfffeedddddddcccccc
bbbccddddddddddddddddeeeeeefffghvasjjqqyqt upqlrjhhhkhfedddddddccccc
bbbcdddddddddddddddeeeeffffffgghks c qnbpfmgfedddddddcccc
bbcdddddddddddddeefffffffffgggipmt qhgfeedddddddccc
bbdddddddddeeeefhlggggggghhhhils ljigfeedddddddcc
bcddddeeeeeefffghmllkjiljjiijle yhfeedddddddcc
bddeeeeeeeffffghhjoj do clmq qlgfeeedddddddc
bdeeeeeefffffhiijpu sm ohffeeedddddddc
beffeefgggghhjocsu higffeeedddddddc
cmihgffeeedddddddd
beffeefgggghhjocsu higffeeedddddddc
bdeeeeeefffffhiijpu sd ohffeeedddddddc
bddeeeeeeeffffghhjoj do clmq qlgfeeedddddddc
bcddddeeeeeefffghmllkjiljjiijle yhfeedddddddcc
bbdddddddddeeeefhlggggggghhhhils ljigfeedddddddcc
bbcdddddddddddddeefffffffffgggipmt qhgfeedddddddccc
bbbcdddddddddddddddeeeeffffffgghks c qnbpfmgfedddddddcccc
bbbccddddddddddddddddeeeeeefffghvasjjqqyqt upqlrjhhhkhfedddddddccccc
bbbbccddddddddddddddddddeeeeeeeffgggghhjq iihgggfffeedddddddcccccc
bbbbbcccddddddddddddddddddeeeeeeeefffggjotx etiigfffeeeeddddddcccccccc
bbbbbbccccddddddddddddddddddeeeeeeeefffghikopjhgffeeeeedddddcccccccccc
bbbbbbbcccccdddddddddddddddddddeeeeeeeffghjpjl feeeeedddddcccccccccccc
Fennel
<lang Fennel>
- !/usr/bin/env fennel
(fn mandelzahl [cr ci max i tr ti tr2 ti2]
"Calculates the Mandelbrot escape number of a complex point c"
(if (>= i max) -1
(>= (+ tr2 ti2) 4) i
(let [(tr ti) (values (+ (- tr2 ti2) cr)
(+ (* tr ti 2) ci))]
(mandelzahl cr ci max (+ i 1)
tr ti (* tr tr) (* ti ti)))))
(fn mandel [w h max]
"Entry point, generate a 'graphical' representation of the Mandelbrot set"
(for [y -1.0 1.0 (/ 2.0 h)]
(var line {})
(for [x -2.0 1.0 (/ 3.0 w)]
(let [mz (mandelzahl x y max 0 0 0 0 0)]
(tset line (+ (length line) 1)
(or (and (< mz 0) " ")
(string.char (+ (string.byte :a) (% mz 26)))))))
(print (table.concat line))))
(fn arg-def [pos default]
"A helper fn to extract command line parameter with defaults" (or (tonumber (and arg (. arg pos))) default))
(let [width (arg-def 1 140)
height (arg-def 2 50)
max (arg-def 3 1e5)]
(mandel width height max))
</lang>
FOCAL
<lang FOCAL>1.1 S I1=-1.2; S I2=1.2; S R1=-2; S R2=.5 1.2 S MIT=30 1.3 F Y=1,24; D 2 1.4 Q
2.1 T ! 2.2 F X=1,70; D 3
3.1 S R=X*(R2-R1)/70+R1 3.2 S I=Y*(I2-I1)/24+I1 3.3 S C1=R; S C2=I 3.4 F T=1,MIT; D 4
4.1 S C3=C1 4.2 S C1=C1*C1 - C2*C2 4.3 S C2=C3*C2 + C2*C3 4.4 S C1=C1+R 4.5 S C2=C2+I 4.6 I (-FABS(C1)+2)5.1 4.7 I (-FABS(C2)+2)5.1 4.8 I (MIT-T-1)6.1
5.1 S T=MIT; T "*"; R
6.1 T " "; R</lang>
- Output:
********************************************************************** ********************************************************************** **************************************************** ***************** ************************************************* *************** ************************************************* *************** **************************************** * ********** *************************************** * ****** ************************************* ***** ********************** **** ******* **** *********************** ** ** ********************* **** *** * ******* ********************* **** *********************** ** ** ********************** **** ******* **** ************************************* ***** *************************************** * ****** **************************************** * ********** ************************************************* *************** ************************************************* *************** **************************************************** ***************** ********************************************************************** ********************************************************************** **********************************************************************
Forth
This uses grayscale image utilities. <lang Forth>500 value max-iter
- mandel ( gmp F: imin imax rmin rmax -- )
0e 0e { F: imin F: imax F: rmin F: rmax F: Zr F: Zi }
dup bheight 0 do
i s>f dup bheight s>f f/ imax imin f- f* imin f+ TO Zi
dup bwidth 0 do
i s>f dup bwidth s>f f/ rmax rmin f- f* rmin f+ TO Zr
Zr Zi max-iter
begin 1- dup
while fover fdup f* fover fdup f*
fover fover f+ 4e f<
while f- Zr f+
frot frot f* 2e f* Zi f+
repeat fdrop fdrop
drop 0 \ for a pretty grayscale image, replace with: 255 max-iter */
else drop 255
then fdrop fdrop
over i j rot g!
loop
loop drop ;
80 24 graymap dup -1e 1e -2e 1e mandel</lang>
This is a completely integer version without local variables, which uses 4tH's native graphics library. <lang Forth>include lib/graphics.4th \ graphics support is needed
640 pic_width ! \ width of the image 480 pic_height ! \ height of the image
create shade \ map the shades of the image
' black , \ this is the colorscheme ' blue , ' cyan , ' green , ' yellow , ' red , ' magenta , ' blue , ' cyan , ' green , ' yellow , ' white ,
does> swap cells + @c execute ; \ loop through the shades available
color_image \ we're making a color image
15121 -15120 do \ do y-coordinate
15481 -21000 do \ do x-coordinate
j 0 0 0 ( l u v i)
200 0 do \ get color
>r
over dup 10 / * 1000 / \ calculate X and Y
over dup 10 / * 1000 / \ if X+Y > 40000
over over + r> swap 40000 > \ use the color in the loop
if
drop drop drop i 11 min leave
else \ otherwise try the next one
j swap >r - - >r * 5000 / over + r> swap r>
then
loop \ drop all parameters and set the shade
shade drop drop drop \ now set the proper pixel
j 15120 + 63 / i 21000 + 57 / set_pixel
57 +loop \ we're scaling the x-coordinate
63 +loop \ we're scaling the y-coordinate
s" mandelbt.ppm" save_image \ done, save the image dup gshow free bye</lang>
Fortran
<lang fortran>program mandelbrot
implicit none integer , parameter :: rk = selected_real_kind (9, 99) integer , parameter :: i_max = 800 integer , parameter :: j_max = 600 integer , parameter :: n_max = 100 real (rk), parameter :: x_centre = -0.5_rk real (rk), parameter :: y_centre = 0.0_rk real (rk), parameter :: width = 4.0_rk real (rk), parameter :: height = 3.0_rk real (rk), parameter :: dx_di = width / i_max real (rk), parameter :: dy_dj = -height / j_max real (rk), parameter :: x_offset = x_centre - 0.5_rk * (i_max + 1) * dx_di real (rk), parameter :: y_offset = y_centre - 0.5_rk * (j_max + 1) * dy_dj integer, dimension (i_max, j_max) :: image integer :: i integer :: j integer :: n real (rk) :: x real (rk) :: y real (rk) :: x_0 real (rk) :: y_0 real (rk) :: x_sqr real (rk) :: y_sqr
do j = 1, j_max
y_0 = y_offset + dy_dj * j
do i = 1, i_max
x_0 = x_offset + dx_di * i
x = 0.0_rk
y = 0.0_rk
n = 0
do
x_sqr = x ** 2
y_sqr = y ** 2
if (x_sqr + y_sqr > 4.0_rk) then
image (i, j) = 255
exit
end if
if (n == n_max) then
image (i, j) = 0
exit
end if
y = y_0 + 2.0_rk * x * y
x = x_0 + x_sqr - y_sqr
n = n + 1
end do
end do
end do
open (10, file = 'out.pgm')
write (10, '(a/ i0, 1x, i0/ i0)') 'P2', i_max, j_max, 255
write (10, '(i0)') image
close (10)
end program mandelbrot</lang>bs
Frink
This draws a graphical Mandelbrot set using Frink's built-in graphics and complex arithmetic. <lang Frink> // Maximum levels for each pixel. levels = 60
// Create a random color for each level. colors = new arraylevels for a = 0 to levels-1
colors@a = new color[randomFloat[0,1], randomFloat[0,1], randomFloat[0,1]]
// Make this number smaller for higher resolution. stepsize = .005
g = new graphics g.antialiased[false]
for im = -1.2 to 1.2 step stepsize {
imag = i * im
for real = -2 to 1 step stepsize
{
C = real + imag
z = 0
count = -1
do
{
z = z^2 + C
count=count+1;
} while abs[z] < 4 and count < levels
g.color[colors@((count-1) mod levels)]
g.fillRectSize[real, im, stepsize, stepsize]
}
}
g.show[] </lang>
Furor
<lang Furor>
- sysinclude X.uh
$ff0000 sto szin 300 sto maxiter maxypixel sto YRES maxxpixel sto XRES myscreen "Mandelbrot" @YRES @XRES graphic @YRES 2 / (#d) sto y2 @YRES 2 / (#d) sto x2
- g 0. @XRES (#d) 1. i: {#d
- g 0. @YRES (#d) 1. {#d
- d
{#d}§i 400. - @x2 - @x2 / sto x {#d} @y2 - @y2 / sto y zero#d xa zero#d ya zero iter (( #d @x @xa dup* @ya dup* -+ @y @xa *2 @ya *+ sto ya sto xa #g inc iter @iter @maxiter >= then((>))
- d ( @xa dup* @ya dup* + 4. > )))
- g @iter @maxiter == { #d
myscreen {d} {d}§i @szin [][] }{ #d myscreen {d} {d}§i #g @iter 64 * [][] }
- d}
- d}
(( ( myscreen key? 10000 usleep ))) myscreen !graphic end { „x” } { „x2” } { „y” } { „y2” } { „xa” } { „ya” } { „iter” } { „maxiter” } { „szin” } { „YRES” } { „XRES” } { „myscreen” } </lang>
Futhark
Computes escapes for each pixel, but not the colour.
<lang Futhark> default(f32)
type complex = (f32, f32)
fun dot(c: complex): f32 =
let (r, i) = c in r * r + i * i
fun multComplex(x: complex, y: complex): complex =
let (a, b) = x
let (c, d) = y
in (a*c - b * d,
a*d + b * c)
fun addComplex(x: complex, y: complex): complex =
let (a, b) = x
let (c, d) = y
in (a + c,
b + d)
fun divergence(depth: int, c0: complex): int =
loop ((c, i) = (c0, 0)) = while i < depth && dot(c) < 4.0 do
(addComplex(c0, multComplex(c, c)),
i + 1)
in i
fun mandelbrot(screenX: int, screenY: int, depth: int, view: (f32,f32,f32,f32)): [screenX][screenY]int =
let (xmin, ymin, xmax, ymax) = view
let sizex = xmax - xmin
let sizey = ymax - ymin
in map (fn (x: int): [screenY]int =>
map (fn (y: int): int =>
let c0 = (xmin + (f32(x) * sizex) / f32(screenX),
ymin + (f32(y) * sizey) / f32(screenY))
in divergence(depth, c0))
(iota screenY))
(iota screenX)
fun main(screenX: int, screenY: int, depth: int, xmin: f32, ymin: f32, xmax: f32, ymax: f32): [screenX][screenY]int =
mandelbrot(screenX, screenY, depth, (xmin, ymin, xmax, ymax))
</lang>
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
GLSL
Uses smooth coloring. <lang glsl> const int MaxIterations = 1000; const vec2 Focus = vec2(-0.51, 0.54); const float Zoom = 1.0;
vec3 color(int iteration, float sqLengthZ) {
// If the point is within the mandlebrot set
// just color it black
if(iteration == MaxIterations)
return vec3(0.0);
// Else we give it a smoothed color
float ratio = (float(iteration) - log2(log2(sqLengthZ))) / float(MaxIterations);
// Procedurally generated colors
return mix(vec3(1.0, 0.0, 0.0), vec3(1.0, 1.0, 0.0), sqrt(ratio));
}
void mainImage(out vec4 fragColor, in vec2 fragCoord) {
// C is the aspect-ratio corrected UV coordinate. vec2 c = (-1.0 + 2.0 * fragCoord / iResolution.xy) * vec2(iResolution.x / iResolution.y, 1.0); // Apply scaling, then offset to get a zoom effect c = (c * exp(-Zoom)) + Focus;
vec2 z = c;
int iteration = 0;
while(iteration < MaxIterations) {
// Precompute for efficiency
float zr2 = z.x * z.x;
float zi2 = z.y * z.y;
// The larger the square length of Z,
// the smoother the shading
if(zr2 + zi2 > 32.0) break;
// Complex multiplication, then addition
z = vec2(zr2 - zi2, 2.0 * z.x * z.y) + c;
++iteration;
}
// Generate the colors
fragColor = vec4(color(iteration, dot(z,z)), 1.0);
// Apply gamma correction
fragColor.rgb = pow(fragColor.rgb, vec3(0.5));
}
</lang>
gnuplot
The output from gnuplot is controlled by setting the appropriate values for the options terminal and output.
<lang gnuplot>set terminal png
set output 'mandelbrot.png'</lang>
The following script draws an image of the number of iterations it takes to escape the circle with radius rmax with a maximum of nmax.
<lang gnuplot>rmax = 2
nmax = 100
complex (x, y) = x * {1, 0} + y * {0, 1}
mandelbrot (z, z0, n) = n == nmax || abs (z) > rmax ? n : mandelbrot (z ** 2 + z0, z0, n + 1)
set samples 200
set isosamples 200
set pm3d map
set size square
splot [-2 : .8] [-1.4 : 1.4] mandelbrot (complex (0, 0), complex (x, y), 0) notitle</lang>
- Output:
Go
- Text
Prints an 80-char by 41-line depiction. <lang go>package main
import "fmt" import "math/cmplx"
func mandelbrot(a complex128) (z complex128) {
for i := 0; i < 50; i++ {
z = z*z + a
}
return
}
func main() {
for y := 1.0; y >= -1.0; y -= 0.05 {
for x := -2.0; x <= 0.5; x += 0.0315 {
if cmplx.Abs(mandelbrot(complex(x, y))) < 2 {
fmt.Print("*")
} else {
fmt.Print(" ")
}
}
fmt.Println("")
}
}</lang>
- Graphical
<lang go>package main
import (
"fmt" "image" "image/color" "image/draw" "image/png" "math/cmplx" "os"
)
const (
maxEsc = 100 rMin = -2. rMax = .5 iMin = -1. iMax = 1. width = 750 red = 230 green = 235 blue = 255
)
func mandelbrot(a complex128) float64 {
i := 0
for z := a; cmplx.Abs(z) < 2 && i < maxEsc; i++ {
z = z*z + a
}
return float64(maxEsc-i) / maxEsc
}
func main() {
scale := width / (rMax - rMin)
height := int(scale * (iMax - iMin))
bounds := image.Rect(0, 0, width, height)
b := image.NewNRGBA(bounds)
draw.Draw(b, bounds, image.NewUniform(color.Black), image.ZP, draw.Src)
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
fEsc := mandelbrot(complex(
float64(x)/scale+rMin,
float64(y)/scale+iMin))
b.Set(x, y, color.NRGBA{uint8(red * fEsc),
uint8(green * fEsc), uint8(blue * fEsc), 255})
}
}
f, err := os.Create("mandelbrot.png")
if err != nil {
fmt.Println(err)
return
}
if err = png.Encode(f, b); err != nil {
fmt.Println(err)
}
if err = f.Close(); err != nil {
fmt.Println(err)
}
}</lang>
Golfscript
Código sacado de https://codegolf.stackexchange.com/ <lang golfscript>20{40{0.1{.{;..*2$.*\- 20/3$-@@*10/3$-..*2$.*+1600<}*}32*'
- '=\;\;@@(}60*;(n\}40*;]+</lang>
- Output:
000000000000000000000000000000000000000010000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000001000000000000000000000000 000000000000000000000000000000000000001000000000000000000000 000000000000000000000000000000000000111000000000000000000000 000000000000000000000000000000000000111110000000000000000000 000000000000000000000000000000000000011100000000000000000000 000000000000000000000000000001000110111100010000000000000000 000000000000000000000000000000100111111111110000000000000000 000000000000000000000000000001011111111111110111000000000000 000000000000000000000000000001111111111111111110000000000000 000000000000000000000000000000111111111111111110000000000000 000000000000001000000000000011111111111111111111000000000000 000000000000000000000000000011111111111111111111000000000000 000000000000000000000000000111111111111111111111000000000000 000000000000000000000000001111111111111111111111100000000000 000000000000000001111110001111111111111111111111100000000000 000000000000000011111111101111111111111111111111100000000000 000000000000100111111111111111111111111111111111000000000000 000000000001101111111111111111111111111111111111000000000000 011111111111111111111111111111111111111111111100000000000000 000000000000001111111111111111111111111111111110000000000000 000000000000000111111111111111111111111111111111000000000000 000000000000000001111111111111111111111111111111100000000000 000000000000000001111111101111111111111111111111000000000000 000000000000000001011100000111111111111111111111100000000000 000000000000000000000100000111111111111111111111000000000000 000000000000000100000000001111111111111111111111100000000000 000000000000000100000000000011111111111111111111000000000000 000000000000000000000000000011111111111111111110000000000000 000000000000000000000000000001111111111111111111000000000000 000000000000000000000000000000111111111111111111000000000000 000000000000000000000000000001101111111111111000000000000000 000000000000000000000000000011000011111110100000000000000000 000000000000000000000000000000000000111100000000000000000000 000000000000000000000000000000000000111110000000000000000000 000000000000000000000000000000000000111100000000000000000000 000000000000000000000000000000000000011000000000000000000000 000000000000000000000000000000000000001000000000000000000000 000000000000000000000000000000000000000000000000000000000000
Hare
<lang hare>use fmt; use math;
type complex = struct { re: f64, im: f64 };
export fn main() void = { for (let y = -1.2; y < 1.2; y += 0.05) { for (let x = -2.05; x < 0.55; x += 0.03) { let z = complex {re = 0.0, im = 0.0};
for (let m = 0z; m < 100; m += 1) { let tz = z;
z.re = tz.re*tz.re - tz.im*tz.im; z.im = tz.re*tz.im + tz.im*tz.re; z.re += x; z.im += y; }; fmt::print(if (abs(z) < 2f64) '#' else '.')!; }; fmt::println()!; }; };
fn abs(z: complex) f64 = { return math::sqrtf64(z.re*z.re + z.im*z.im); };</lang>
- Output:
....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ................................................................##..................... .............................................................######.................... .............................................................#######................... ..............................................................######................... ..........................................................#.#.###..#.#................. ...................................................##....################.............. ..................................................###.######################.###....... ...................................................############################........ ................................................###############################........ ................................................################################....... .............................................#####################################..... ..............................................###################################...... ..............................##.####.#......####################################...... ..............................###########....####################################...... ............................###############.######################################..... ............................###############.#####################################...... ........................##.#####################################################....... ......#.###...#.#############################################################.......... ........................##.#####################################################....... ............................###############.#####################################...... ............................###############.######################################..... ..............................###########....####################################...... ..............................##.####.#......####################################...... ..............................................###################################...... .............................................#####################################..... ................................................################################....... ................................................###############################........ ...................................................############################........ ..................................................###.######################.###....... ...................................................##....################.............. ..........................................................#.#.###..#.#................. ..............................................................######................... .............................................................#######................... .............................................................######.................... ................................................................##..................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... ....................................................................................... .......................................................................................
Haskell
<lang haskell>import Data.Bool import Data.Complex (Complex((:+)), magnitude)
mandelbrot
:: RealFloat a => Complex a -> Complex a
mandelbrot a = iterate ((a +) . (^ 2)) 0 !! 50
main :: IO () main =
mapM_
putStrLn
[ [ bool ' ' '*' (2 > magnitude (mandelbrot (x :+ y)))
| x <- [-2,-1.9685 .. 0.5] ]
| y <- [1,0.95 .. -1] ]</lang>
Save the code to file m.hs and run :
runhaskell m.hs
- Output:
**
******
********
******
******** ** *
*** *****************
************************ ***
****************************
******************************
******************************
************************************
* **********************************
** ***** * **********************************
*********** ************************************
************** ************************************
***************************************************
*****************************************************
***********************************************************************
*****************************************************
***************************************************
************** ************************************
*********** ************************************
** ***** * **********************************
* **********************************
************************************
******************************
******************************
****************************
************************ ***
*** *****************
******** ** *
******
********
******
**
haskell one-liners : <lang haskell>
-- first attempt -- putStrLn $ foldr (++) "" [ if x==(-2) then "\n" else let (a, b) = iterate (\(x', y') -> (x'^2-y'^2+x, 2*x'*y'+y)) (0, 0) !! 500 in (snd.head.filter (\(v, c)->v) $ zip ([(<0.01), (<0.025), (<0.05), (<0.1), (<0.5), (<1), (<4), (\_ -> True)] <*> [a^2 + b^2]) [".", "\'", ":", "!", "|", "}", "#", " "]) | y <- [1, 0.98 .. -1], x <- [-2, -1.98 .. 0.5]]
-- replaced iterate with foldr, modified the snd.head part and a introduced a check to stop the magnitude from exploding -- foldr(>>)(return())[putStrLn[let(a,b)=foldr(\_(u,w)->if(u^2+w^2<4)then(u^2-w^2+x,2*u*w+y)else(u,w))(0,0)[1..500]in snd.last$(filter(\(f,v)->f)$zip(map(a^2+b^2>)[0,0.01,0.025,0.05,0.1,0.5,1,4])['.','\,':','!','|','}','#',' '])|x<-[-2,-1.98..0.5]]|y<-[1,0.98.. -1]]
-- without different characters in the output -- foldr(>>)(return())[putStrLn[let(a,b)=foldr(\_(u,w)->(u^2-w^2+x,2*u*w+y))(0,0)[1..500]in if a^2+b^2<4 then '*' else ' '|x<-[-2,-1.98..0.5]]|y<-[1,0.98.. -1]]
-- using mapM_ instead of foldr, bind operator instead of list comprehension and replacing 'let' with a lambda function
-- mapM_ putStrLn $[1,0.98.. -1]>>= \y->return $[-2,-1.98..0.5]>>= \x->return (if(\(a,b)->a^2+b^2<4)(foldr(\_(u,w)->(u^2-w^2+x,2*u*w+y))(0,0)[1..500]) then '*' else ' ')
-- open GHCI > Copy and paste any of above one-liners > Hit enter </lang>
A legible variant of the first of the "one-liner" contributions above:
<lang haskell>main :: IO () main =
putStrLn $
concat
[ go x y
| y <- [1, 0.98 .. -1],
x <- [-2, -1.98 .. 0.5]
]
where
go x y
| x == (-2) = "\n"
| otherwise =
let (a, b) =
iterate
(\(x', y') -> (x' ^ 2 - y' ^ 2 + x, 2 * x' * y' + y))
(0, 0)
!! 500
in ( snd . head . filter fst $
zip
( [ (< 0.01),
(< 0.025),
(< 0.05),
(< 0.1),
(< 0.5),
(< 1),
(< 4),
const True
]
<*> [a ^ 2 + b ^ 2]
)
[".", "\'", ":", "!", "|", "}", "#", " "]
)</lang>
- Output:
#
##
#
}}}}}}}
#}}}}}}}}
}}}}}}}}}
}}}}}}}}}
}}}}}}}}}#
}}}}}}}}}}
}}}}}}}}
}}}}}}}
|||}}
} |||||||||||
} ||||||||||||||||||| |
. ' |||||||||||||||||||||||||
..' ||||||||||||||||||||||||||||
'... ||||||||||||||||||||||||||||||| '.':
'':|||||||||||||||||||||||||||||||||| :...'
||||||||||||||||||||||||||||||||||||:'..'
|||||||||||||||||||||||||||||||||||||||:''
}|||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||
|| ||||||||||||||||||||||||||||||||||||||||||||!
||}|||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||!!!!!|||||||||||||||||| :
|||||||||||||||||||||||||!!!!!!!!!!!!|||||||||||||||'.'
||||||||||||||||||||||!!!!!!!!!!!!!!!!||||||||||||| .
.||||||||||||||||||||!!!!!!!!!!!!!!!!!!!!||||||||||||
| |||||||||||||||||||||!!!!!!!!!!!!!!!!!!!!!!|||||||||||
| | |||||||||||||||||||!!!!!!!!!!!!!!!!!!!!!!!!||||||||||
||! .:::!!!!. ||||||||||||||||||||!!!!!!!!!!!!!:::!!!!!!!!!||||||||||
!'::::::::!!! ||||||||||||||||||!!!!!!!!!!!:::::::::!!!!!!!|||||||||||
::::::::::::!!!| |||||||||||||||||||!!!!!!!!!:::::::::::::!!!!!!||||||||
!::::::::::::::!!!! ||||||||||||||||||!!!!!!!!!:::::::::::::::!!!!!||||||||
. ::::::''''::::::!!! |||||||||||||||||!!!!!!!!!:::::::''''::::::!!!!||||||||
:::::'''''''''::::!!! |||||||||||||||||!!!!!!!!::::::'''''''''::::!!!!||||||||
!::::''''''''''':::!!! ||||||||||||||||||!!!!!!!::::::''''''''''':::!!!!||||||!
::::''''''..''''':::!!! |||||||||||||||||!!!!!!!!:::::''''''..''''':::!!!||||||}
:::'''''......''':::!!! |||||||||||||||||!!!!!!!:::::'''''......''':::!!!||||||
::::''''........'''::!!!||||||||||||||||||!!!!!!!:::::''''........'''::!!!|||||
.... ::::'''..........''::!!!||||||||||||||||||!!!!!!!:::::'''..........''::!!||||||
.....':::''''..........''::!!|||||||||||||||||||!!!!!!!::::''''..........''::!!||||
## | ::.....':::''''..........''::!!|||||||||||||||||||!!!!!!!::::''''..........''::!!||
.....':::''''..........''::!!|||||||||||||||||||!!!!!!!::::''''..........''::!!||||
.... ::::'''..........''::!!!||||||||||||||||||!!!!!!!:::::'''..........''::!!||||||
::::''''........'''::!!!||||||||||||||||||!!!!!!!:::::''''........'''::!!!|||||
:::'''''......''':::!!! |||||||||||||||||!!!!!!!:::::'''''......''':::!!!||||||
::::''''''..''''':::!!! |||||||||||||||||!!!!!!!!:::::''''''..''''':::!!!||||||}
!::::''''''''''':::!!! ||||||||||||||||||!!!!!!!::::::''''''''''':::!!!!||||||!
:::::'''''''''::::!!! |||||||||||||||||!!!!!!!!::::::'''''''''::::!!!!||||||||
. ::::::''''::::::!!! |||||||||||||||||!!!!!!!!!:::::::''''::::::!!!!||||||||
!::::::::::::::!!!! ||||||||||||||||||!!!!!!!!!:::::::::::::::!!!!!||||||||
::::::::::::!!!| |||||||||||||||||||!!!!!!!!!:::::::::::::!!!!!!||||||||
!'::::::::!!! ||||||||||||||||||!!!!!!!!!!!:::::::::!!!!!!!|||||||||||
||! .:::!!!!. ||||||||||||||||||||!!!!!!!!!!!!!:::!!!!!!!!!||||||||||
| | |||||||||||||||||||!!!!!!!!!!!!!!!!!!!!!!!!||||||||||
| |||||||||||||||||||||!!!!!!!!!!!!!!!!!!!!!!|||||||||||
.||||||||||||||||||||!!!!!!!!!!!!!!!!!!!!||||||||||||
||||||||||||||||||||||!!!!!!!!!!!!!!!!||||||||||||| .
|||||||||||||||||||||||||!!!!!!!!!!!!|||||||||||||||'.'
|||||||||||||||||||||||||||!!!!||||||||||||||||||| :
|||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||
||}|||||||||||||||||||||||||||||||||||||||||||||
|| ||||||||||||||||||||||||||||||||||||||||||||!
|||||||||||||||||||||||||||||||||||||||||||
}|||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||:''
||||||||||||||||||||||||||||||||||||:'..'
'':|||||||||||||||||||||||||||||||||| :...'
'... ||||||||||||||||||||||||||||||| '.':
..' ||||||||||||||||||||||||||||
. ' |||||||||||||||||||||||||
} ||||||||||||||||||| |
} |||||||||||
|||}}
}}}}}}}
}}}}}}}}
}}}}}}}}}}
}}}}}}}}}#
}}}}}}}}}
}}}}}}}}}
#}}}}}}}}
}}}}}}}
#
##
#
and a legible variant of the last of the "one-liner" contributions above: <lang haskell>main :: IO () main =
mapM_
putStrLn
$ [1, 0.98 .. -1]
>>= \y ->
[ [-2, -1.98 .. 0.5]
>>= \x ->
[ if (\(a, b) -> a ^ 2 + b ^ 2 < 4)
( foldr
( \_ (u, w) ->
(u ^ 2 - w ^ 2 + x, 2 * u * w + y)
)
(0, 0)
[1 .. 500]
)
then '*'
else ' '
]
]</lang>
Haxe
This version compiles for flash version 9 or greater. The compilation command is <lang haxe>haxe -swf mandelbrot.swf -main Mandelbrot</lang>
<lang haxe>class Mandelbrot extends flash.display.Sprite {
inline static var MAX_ITER = 255;
public static function main() {
var w = flash.Lib.current.stage.stageWidth;
var h = flash.Lib.current.stage.stageHeight;
var mandelbrot = new Mandelbrot(w, h);
flash.Lib.current.stage.addChild(mandelbrot);
mandelbrot.drawMandelbrot();
}
var image:flash.display.BitmapData;
public function new(width, height) {
super();
var bitmap:flash.display.Bitmap;
image = new flash.display.BitmapData(width, height, false);
bitmap = new flash.display.Bitmap(image);
this.addChild(bitmap);
}
public function drawMandelbrot() {
image.lock();
var step_x = 3.0 / (image.width-1);
var step_y = 2.0 / (image.height-1);
for (i in 0...image.height) {
var ci = i * step_y - 1.0;
for (j in 0...image.width) {
var k = 0;
var zr = 0.0;
var zi = 0.0;
var cr = j * step_x - 2.0;
while (k <= MAX_ITER && (zr*zr + zi*zi) <= 4) {
var temp = zr*zr - zi*zi + cr;
zi = 2*zr*zi + ci;
zr = temp;
k ++;
}
paint(j, i, k);
}
}
image.unlock();
}
inline function paint(x, y, iter) {
var color = iter > MAX_ITER? 0 : iter * 0x100;
image.setPixel(x, y, color);
}
}</lang>
Huginn
<lang huginn>#! /bin/sh exec huginn -E "${0}" "${@}"
- ! huginn
import Algorithms as algo; import Mathematics as math; import Terminal as term;
mandelbrot( x, y ) {
c = math.Complex( x, y );
z = math.Complex( 0., 0. );
s = -1;
for ( i : algo.range( 50 ) ) {
z = z * z + c;
if ( | z | > 2. ) {
s = i;
break;
}
}
return ( s );
}
main( argv_ ) {
imgSize = term_size( argv_ );
yRad = 1.2;
yScale = 2. * yRad / real( imgSize[0] );
xScale = 3.3 / real( imgSize[1] );
glyphTab = [ ".", ":", "-", "+", "+" ].resize( 12, "*" ).resize( 26, "%" ).resize( 50, "@" ).push( " " );
for ( y : algo.range( imgSize[0] ) ) {
line = "";
for ( x : algo.range( imgSize[1] ) ) {
line += glyphTab[ mandelbrot( xScale * real( x ) - 2.3, yScale * real( y ) - yRad ) ];
}
print( line + "\n" );
}
return ( 0 );
}
term_size( argv_ ) {
lines = 25;
columns = 80;
if ( size( argv_ ) == 3 ) {
lines = integer( argv_[1] );
columns = integer( argv_[2] );
} else {
lines = term.lines();
columns = term.columns();
if ( ( lines % 2 ) == 0 ) {
lines -= 1;
}
}
lines -= 1;
columns -= 1;
return ( ( lines, columns ) );
}</lang>
- Output:
........................:::::::::::::::::::------------------------------------------------------:::::::::::::::::::::::::::::::::::: ......................::::::::::::::::------------------------------------++++++++++++++++++++---------:::::::::::::::::::::::::::::: ....................:::::::::::::-----------------------------------+++++++++++++*******+++++++++++--------:::::::::::::::::::::::::: ...................:::::::::::----------------------------------++++++++++++++++****%%******++++++++++---------:::::::::::::::::::::: .................:::::::::----------------------------------++++++++++++++++++++******% %****++++++++++++---------::::::::::::::::::: ................::::::::---------------------------------+++++++++++++++++++++******%%%%%*****+++++++++++++----------:::::::::::::::: ...............::::::---------------------------------++++++++++++++++++++*****%%%%% @%%%@***++++++++++++++----------:::::::::::::: ..............:::::--------------------------------++++++++++++++++++***********@% @%******+++++++++++++-----------::::::::::: .............::::-------------------------------+++++++++++++++++****************@ %%***************+++++------------::::::::: ............:::------------------------------+++++++++++++++++****@%%%****%@@%@% %%@ @%%%% %*% ********%**++++------------:::::::: ...........:::----------------------------+++++++++++++++++********% @%@@ %%%**%@%%@@**+++++------------:::::: ..........:::-------------------------+++++++++++++++++++**********% @ %%***++++++------------::::: ..........:----------------------+++++++++++++++++++++*********%%%% %%******++++++------------:::: .........:-----------------++++++++**************************** %*****+++++-------------::: .........----------+++++++++++++++****%********%%*********** @%% % %%**++++++-------------:: ........:------++++++++++++++++++*******%@@%%**%% %%%*******%% %%***+++++++-------------: ........---+++++++++++++++++++++********%@ @%%%**% %%***+++++++-------------: .......:-+++++++++++++++++++++*******%%%% @%% @****++++++++------------- .......-+++++++++++++***********%**@*%@ % %***+++++++++------------- .......++++++*******************%% @ %*****+++++++++------------- ....... %%******++++++++++------------- .......++++++*******************%% @ %*****+++++++++------------- .......-+++++++++++++***********%**@*%@ % %***+++++++++------------- .......:-+++++++++++++++++++++*******%%%% @%% @****++++++++------------- ........---+++++++++++++++++++++********%@ @%%%**% %%***+++++++-------------: ........:------++++++++++++++++++*******%@@%%**%% %%%*******%% %%***+++++++-------------: .........----------+++++++++++++++****%********%%*********** @%% % %%**++++++-------------:: .........:-----------------++++++++**************************** %*****+++++-------------::: ..........:----------------------+++++++++++++++++++++*********%%%% %%******++++++------------:::: ..........:::-------------------------+++++++++++++++++++**********% @ %%***++++++------------::::: ...........:::----------------------------+++++++++++++++++********% @%@@ %%%**%@%%@@**+++++------------:::::: ............:::------------------------------+++++++++++++++++****@%%%****%@@%@% %%@ @%%%% %*% ********%**++++------------:::::::: .............::::-------------------------------+++++++++++++++++****************@ %%***************+++++------------::::::::: ..............:::::--------------------------------++++++++++++++++++***********@% @%******+++++++++++++-----------::::::::::: ...............::::::---------------------------------++++++++++++++++++++*****%%%%% @%%%@***++++++++++++++----------:::::::::::::: ................::::::::---------------------------------+++++++++++++++++++++******%%%%%*****+++++++++++++----------:::::::::::::::: .................:::::::::----------------------------------++++++++++++++++++++******% %****++++++++++++---------::::::::::::::::::: ...................:::::::::::----------------------------------++++++++++++++++****%%******++++++++++---------:::::::::::::::::::::: ....................:::::::::::::-----------------------------------+++++++++++++*******+++++++++++--------:::::::::::::::::::::::::: ......................::::::::::::::::------------------------------------++++++++++++++++++++---------::::::::::::::::::::::::::::::
Icon and Unicon
<lang Icon>link graphics
procedure main()
width := 750
height := 600
limit := 100
WOpen("size="||width||","||height)
every x:=1 to width & y:=1 to height do
{
z:=complex(0,0)
c:=complex(2.5*x/width-2.0,(2.0*y/height-1.0))
j:=0
while j<limit & cAbs(z)<2.0 do
{
z := cAdd(cMul(z,z),c)
j+:= 1
}
Fg(mColor(j,limit))
DrawPoint(x,y)
}
WriteImage("./mandelbrot.gif")
WDone()
end
procedure mColor(x,limit)
max_color := 2^16-1 color := integer(max_color*(real(x)/limit))
return(if x=limit
then "black"
else color||","||color||",0")
end
record complex(r,i)
procedure cAdd(x,y)
return complex(x.r+y.r,x.i+y.i)
end
procedure cMul(x,y)
return complex(x.r*y.r-x.i*y.i,x.r*y.i+x.i*y.r)
end
procedure cAbs(x)
return sqrt(x.r*x.r+x.i*x.i)
end</lang>
IDL
IDL - Interactive Data Language (free implementation: GDL - GNU Data Language http://gnudatalanguage.sourceforge.net) <lang IDL> PRO Mandelbrot,xRange,yRange,xPixels,yPixels,iterations
xPixelstartVec = Lindgen( xPixels) * Float(xRange[1]-xRange[0]) / $
xPixels + xRange[0]
yPixelstartVec = Lindgen( yPixels) * Float(YRANGE[1]-yrange[0])$
/ yPixels + yRange[0]
constArr = Complex( Rebin( xPixelstartVec, xPixels, yPixels),$
Rebin( Transpose(yPixelstartVec), xPixels, yPixels))
valArr = ComplexArr( xPixels, yPixels)
res = IntArr( xPixels, yPixels)
oriIndex = Lindgen( Long(xPixels) * yPixels)
FOR i = 0, iterations-1 DO BEGIN ; only one loop needed
; calculation for whole array at once valArr = valArr^2 - constArr
whereIn = Where( Abs( valArr) LE 4.0d, COMPLEMENT=whereOut)
IF whereIn[0] EQ -1 THEN BREAK
valArr = valArr[ whereIn]
constArr = constArr[ whereIn]
IF whereOut[0] NE -1 THEN BEGIN
res[ oriIndex[ whereOut]] = i+1
oriIndex = oriIndex[ whereIn] ENDIF
ENDFOR
tv,res ; open a window and show the result
END
Mandelbrot,[-1.,2.3],[-1.3,1.3],640,512,200
END
</lang> from the command line: <lang IDL> GDL>.run mandelbrot </lang> or <lang IDL> GDL> Mandelbrot,[-1.,2.3],[-1.3,1.3],640,512,200 </lang>
Inform 7
<lang inform7>"Mandelbrot"
The story headline is "A Non-Interactive Set".
Include Glimmr Drawing Commands by Erik Temple.
[Q20 fixed-point or floating-point: see definitions below] Use floating-point math.
Finished is a room.
The graphics-window is a graphics g-window spawned by the main-window. The position is g-placeabove.
When play begins: let f10 be 10 as float; now min re is ( -20 as float ) fdiv f10; now max re is ( 6 as float ) fdiv f10; now min im is ( -12 as float ) fdiv f10; now max im is ( 12 as float ) fdiv f10; now max iterations is 100; add color g-Black to the palette; add color g-Red to the palette; add hex "#FFA500" to the palette; add color g-Yellow to the palette; add color g-Green to the palette; add color g-Blue to the palette; add hex "#4B0082" to the palette; add hex "#EE82EE" to the palette; open up the graphics-window.
Min Re is a number that varies. Max Re is a number that varies. Min Im is a number that varies. Max Im is a number that varies.
Max Iterations is a number that varies.
Min X is a number that varies. Max X is a number that varies. Min Y is a number that varies. Max Y is a number that varies.
The palette is a list of numbers that varies.
[vertically mirrored version] Window-drawing rule for the graphics-window when max im is fneg min im: clear the graphics-window; let point be { 0, 0 }; now min X is 0 as float; now min Y is 0 as float; let mX be the width of the graphics-window minus 1; let mY be the height of the graphics-window minus 1; now max X is mX as float; now max Y is mY as float; let L be the column order with max mX; repeat with X running through L: now entry 1 in point is X; repeat with Y running from 0 to mY / 2: now entry 2 in point is Y; let the scaled point be the complex number corresponding to the point; let V be the Mandelbrot result for the scaled point; let C be the color corresponding to V; if C is 0, next; draw a rectangle (C) in the graphics-window at the point with size 1 by 1; now entry 2 in point is mY - Y; draw a rectangle (C) in the graphics-window at the point with size 1 by 1; yield to VM; rule succeeds.
[slower non-mirrored version] Window-drawing rule for the graphics-window: clear the graphics-window; let point be { 0, 0 }; now min X is 0 as float; now min Y is 0 as float; let mX be the width of the graphics-window minus 1; let mY be the height of the graphics-window minus 1; now max X is mX as float; now max Y is mY as float; let L be the column order with max mX; repeat with X running through L: now entry 1 in point is X; repeat with Y running from 0 to mY: now entry 2 in point is Y; let the scaled point be the complex number corresponding to the point; let V be the Mandelbrot result for the scaled point; let C be the color corresponding to V; if C is 0, next; draw a rectangle (C) in the graphics-window at the point with size 1 by 1; yield to VM; rule succeeds.
To decide which list of numbers is column order with max (N - number): let L be a list of numbers; let L2 be a list of numbers; let D be 64; let rev be false; while D > 0: let X be 0; truncate L2 to 0 entries; while X <= N: if D is 64 or X / D is odd, add X to L2; increase X by D; if rev is true: reverse L2; let rev be false; otherwise: let rev be true; add L2 to L; let D be D / 2; decide on L.
To decide which list of numbers is complex number corresponding to (P - list of numbers): let R be a list of numbers; extend R to 2 entries; let X be entry 1 in P as float; let X be (max re fsub min re) fmul (X fdiv max X); let X be X fadd min re; let Y be entry 2 in P as float; let Y be (max im fsub min im) fmul (Y fdiv max Y); let Y be Y fadd min im; now entry 1 in R is X; now entry 2 in R is Y; decide on R.
To decide which number is Mandelbrot result for (P - list of numbers): let c_re be entry 1 in P; let c_im be entry 2 in P; let z_re be 0 as float; let z_im be z_re; let threshold be 4 as float; let runs be 0; while 1 is 1: [ z = z * z ] let r2 be z_re fmul z_re; let i2 be z_im fmul z_im; let ri be z_re fmul z_im; let z_re be r2 fsub i2; let z_im be ri fadd ri; [ z = z + c ] let z_re be z_re fadd c_re; let z_im be z_im fadd c_im; let norm be (z_re fmul z_re) fadd (z_im fmul z_im); increase runs by 1; if norm is greater than threshold, decide on runs; if runs is max iterations, decide on 0.
To decide which number is color corresponding to (V - number): let L be the number of entries in the palette; let N be the remainder after dividing V by L; decide on entry (N + 1) in the palette.
Section - Fractional numbers (for Glulx only)
To decide which number is (N - number) as float: (- (numtof({N})) -). To decide which number is (N - number) fadd (M - number): (- (fadd({N}, {M})) -). To decide which number is (N - number) fsub (M - number): (- (fsub({N}, {M})) -). To decide which number is (N - number) fmul (M - number): (- (fmul({N}, {M})) -). To decide which number is (N - number) fdiv (M - number): (- (fdiv({N}, {M})) -). To decide which number is fneg (N - number): (- (fneg({N})) -). To yield to VM: (- glk_select_poll(gg_event); -).
Use Q20 fixed-point math translates as (- Constant Q20_MATH; -). Use floating-point math translates as (- Constant FLOAT_MATH; -).
Include (-
- ifdef Q20_MATH;
! Q11.20 format: 1 sign bit, 11 integer bits, 20 fraction bits [ numtof n r; @shiftl n 20 r; return r; ]; [ fadd n m; return n+m; ]; [ fsub n m; return n-m; ]; [ fmul n m; n = n + $$1000000000; @sshiftr n 10 n; m = m + $$1000000000; @sshiftr m 10 m; return n * m; ]; [ fdiv n m; @sshiftr m 20 m; return n / m; ]; [ fneg n; return -n; ];
- endif;
- ifdef FLOAT_MATH;
[ numtof f; @"S2:400" f f; return f; ]; [ fadd n m; @"S3:416" n m n; return n; ]; [ fsub n m; @"S3:417" n m n; return n; ]; [ fmul n m; @"S3:418" n m n; return n; ]; [ fdiv n m; @"S3:419" n m n; return n; ]; [ fneg n; @bitxor n $80000000 n; return n; ];
- endif;
-).</lang>
Newer Glulx interpreters provide 32-bit floating-point operations, but this solution also supports fixed-point math which is more widely supported and accurate enough for a zoomed-out view. Inform 6 inclusions are used for the low-level math functions in either case. The rendering process is extremely slow, since the graphics system is not optimized for pixel-by-pixel drawing, so this solution includes an optimization for vertical symmetry (as in the default view) and also includes extra logic to draw the lines in a more immediately useful order.
J
The characteristic function of the Mandelbrot can be defined as follows: <lang j>mcf=. (<: 2:)@|@(] ((*:@] + [)^:((<: 2:)@|@])^:1000) 0:) NB. 1000 iterations test</lang> The Mandelbrot set can be drawn as follows: <lang j>domain=. |.@|:@({.@[ + ] *~ j./&i.&>/@+.@(1j1 + ] %~ -~/@[))&>/
load 'viewmat' viewmat mcf "0 @ domain (_2j_1 1j1) ; 0.01 NB. Complex interval and resolution</lang>
A smaller version, based on a black&white implementation of viewmat (and paraphrased, from html markup to wiki markup), is shown here:
<lang j> viewmat mcf "0 @ domain (_2j_1 1j1) ; 0.1 NB. Complex interval and resolution</lang>
The output is HTML-heavy and can be found here (split out to make editing this page easier).
Java
<lang java>import java.awt.Graphics; import java.awt.image.BufferedImage; import javax.swing.JFrame;
public class Mandelbrot extends JFrame {
private final int MAX_ITER = 570; private final double ZOOM = 150; private BufferedImage I; private double zx, zy, cX, cY, tmp;
public Mandelbrot() {
super("Mandelbrot Set");
setBounds(100, 100, 800, 600);
setResizable(false);
setDefaultCloseOperation(EXIT_ON_CLOSE);
I = new BufferedImage(getWidth(), getHeight(), BufferedImage.TYPE_INT_RGB);
for (int y = 0; y < getHeight(); y++) {
for (int x = 0; x < getWidth(); x++) {
zx = zy = 0;
cX = (x - 400) / ZOOM;
cY = (y - 300) / ZOOM;
int iter = MAX_ITER;
while (zx * zx + zy * zy < 4 && iter > 0) {
tmp = zx * zx - zy * zy + cX;
zy = 2.0 * zx * zy + cY;
zx = tmp;
iter--;
}
I.setRGB(x, y, iter | (iter << 8));
}
}
}
@Override
public void paint(Graphics g) {
g.drawImage(I, 0, 0, this);
}
public static void main(String[] args) {
new Mandelbrot().setVisible(true);
}
}</lang>
Interactive
<lang java>import static java.awt.Color.HSBtoRGB; import static java.awt.Color.black; import static java.awt.event.KeyEvent.VK_BACK_SLASH; import static java.awt.event.KeyEvent.VK_ESCAPE; import static java.awt.image.BufferedImage.TYPE_INT_RGB; import static java.lang.Integer.signum; import static java.lang.Math.abs; import static java.lang.Math.max; import static java.lang.Math.min; import static java.lang.System.currentTimeMillis; import static java.util.Locale.ROOT;
import java.awt.Dimension; import java.awt.Graphics; import java.awt.Insets; import java.awt.event.KeyAdapter; import java.awt.event.KeyEvent; import java.awt.event.MouseAdapter; import java.awt.event.MouseEvent; import java.awt.image.BufferedImage; import java.util.function.Consumer; import java.util.function.Predicate;
import javax.swing.JFrame;
/*
* click: point to center * ctrl-click: point to origin * drag: point to mouse release point * ctrl-drag: point to origin + zoom * back-slash: back to previous point * esc: back to previous zoom point - zoom */
public class Mandelbrot extends JFrame { private static final long serialVersionUID = 1L;
private Insets insets; private int width, height; private double widthHeightRatio; private int minX, minY; private double Zoom;
private int mpX, mpY, mdX, mdY; private boolean isCtrlDown, ctrl; private Stack stack = new Stack();
private BufferedImage image; private boolean newImage = true;
public static void main(String[] args) { new Mandelbrot(800, 600); // (800, 600), (1024, 768), (1600, 900), (1920, 1080) //new Mandelbrot(800, 600, 4.5876514379235943e-09, -0.6092161175392330, -0.4525577210859453); //new Mandelbrot(800, 600, 5.9512354925205320e-10, -0.6092146769531246, -0.4525564820098262); //new Mandelbrot(800, 600, 6.7178527589983420e-08, -0.7819036465400592, -0.1298363433443265); //new Mandelbrot(800, 600, 4.9716091454775210e-09, -0.7818800036717134, -0.1298044093748981); //new Mandelbrot(800, 600, 7.9333341281639390e-06, -0.7238770725243187, -0.2321214232559487); /* new Mandelbrot(800, 600, new double[][] { {5.0000000000000000e-03, -2.6100000000000000, -1.4350000000000000}, // done! {3.5894206549118390e-04, -0.7397795969773300, -0.4996473551637279}, // done! {3.3905106941862460e-05, -0.6270410477828043, -0.4587021918164572}, // done! {6.0636337351945460e-06, -0.6101531446039512, -0.4522561221394852}, // done! {1.5502741161769430e-06, -0.6077214060084073, -0.4503995886987711}, // done! }); //*/ }
public Mandelbrot(int width, int height) { this(width, height, .005, -2.61, -1.435); }
public Mandelbrot(int width, int height, double Zoom, double r, double i) { this(width, height, new double[] {Zoom, r, i}); }
public Mandelbrot(int width, int height, double[] ... points) { super("Mandelbrot Set"); setResizable(false); setDefaultCloseOperation(EXIT_ON_CLOSE); Dimension screen = getToolkit().getScreenSize(); setBounds( rint((screen.getWidth() - width) / 2), rint((screen.getHeight() - height) / 2), width, height ); addMouseListener(mouseAdapter); addMouseMotionListener(mouseAdapter); addKeyListener(keyAdapter); Point point = stack.push(points); this.Zoom = point.Zoom; this.minX = point.minX; this.minY = point.minY; setVisible(true); insets = getInsets(); this.width = width -= insets.left + insets.right; this.height = height -= insets.top + insets.bottom; widthHeightRatio = (double) width / height; }
private int rint(double d) { return (int) Math.rint(d); // half even }
private void repaint(boolean newImage) { this.newImage = newImage; repaint(); }
private MouseAdapter mouseAdapter = new MouseAdapter() { public void mouseClicked(MouseEvent e) { stack.push(false); if (!ctrl) { minX -= width / 2 ; minY -= height / 2; } minX += e.getX() - insets.left; minY += e.getY() - insets.top; ctrl = false; repaint(true); } public void mousePressed(MouseEvent e) { mpX = e.getX(); mpY = e.getY(); ctrl = isCtrlDown; } public void mouseDragged(MouseEvent e) { if (!ctrl) return; setMdCoord(e); repaint(); } private void setMdCoord(MouseEvent e) { int dx = e.getX() - mpX; int dy = e.getY() - mpY; mdX = mpX + max(abs(dx), rint(abs(dy) * widthHeightRatio) * signum(dx)); mdY = mpY + max(abs(dy), rint(abs(dx) / widthHeightRatio) * signum(dy)); acceptIf(insets.left, ge(mdX), setMdXY); acceptIf(insets.top, ge(mdY), setMdYX); acceptIf(insets.left+width-1, le(mdX), setMdXY); acceptIf(insets.top+height-1, le(mdY), setMdYX); } private void acceptIf(int value, Predicate<Integer> p, Consumer<Integer> c) { if (p.test(value)) c.accept(value); } private Predicate<Integer> ge(int md) { return v-> v >= md; } private Predicate<Integer> le(int md) { return v-> v <= md; } private Consumer<Integer> setMdXY = v-> mdY = mpY + rint(abs((mdX=v)-mpX) / widthHeightRatio) * signum(mdY-mpY); private Consumer<Integer> setMdYX = v-> mdX = mpX + rint(abs((mdY=v)-mpY) * widthHeightRatio) * signum(mdX-mpX); public void mouseReleased(MouseEvent e) { if (e.getX() == mpX && e.getY() == mpY) return; stack.push(ctrl); if (!ctrl) { minX += mpX - (mdX = e.getX()); minY += mpY - (mdY = e.getY()); } else { setMdCoord(e); if (mdX < mpX) { int t=mpX; mpX=mdX; mdX=t; } if (mdY < mpY) { int t=mpY; mpY=mdY; mdY=t; } minX += mpX - insets.left; minY += mpY - insets.top; double rZoom = (double) width / abs(mdX - mpX); minX *= rZoom; minY *= rZoom; Zoom /= rZoom; } ctrl = false; repaint(true); } };
private KeyAdapter keyAdapter = new KeyAdapter() { public void keyPressed(KeyEvent e) { isCtrlDown = e.isControlDown(); } public void keyReleased(KeyEvent e) { isCtrlDown = e.isControlDown(); } public void keyTyped(KeyEvent e) { char c = e.getKeyChar(); boolean isEsc = c == VK_ESCAPE; if (!isEsc && c != VK_BACK_SLASH) return; repaint(stack.pop(isEsc)); } };
private class Point { public boolean type; public double Zoom; public int minX; public int minY; Point(boolean type, double Zoom, int minX, int minY) { this.type = type; this.Zoom = Zoom; this.minX = minX; this.minY = minY; } } private class Stack extends java.util.Stack<Point> { private static final long serialVersionUID = 1L; public Point push(boolean type) { return push(type, Zoom, minX, minY); } public Point push(boolean type, double ... point) { double Zoom = point[0]; return push(type, Zoom, rint(point[1]/Zoom), rint(point[2]/Zoom)); } public Point push(boolean type, double Zoom, int minX, int minY) { return push(new Point(type, Zoom, minX, minY)); } public Point push(double[] ... points) { Point lastPoint = push(false, points[0]); for (int i=0, e=points.length-1; i<e; i+=1) { double[] point = points[i]; lastPoint = push(point[0] != points[i+1][0], point); done(printPoint(lastPoint)); } return lastPoint; } public boolean pop(boolean type) { for (;;) { if (empty()) return false; Point d = super.pop(); Zoom = d.Zoom; minX = d.minX; minY = d.minY; if (!type || d.type) return true; } } }
@Override public void paint(Graphics g) { if (newImage) newImage(); g.drawImage(image, insets.left, insets.top, this); //g.drawLine(insets.left+width/2, insets.top+0, insets.left+width/2, insets.top+height); //g.drawLine(insets.left+0, insets.top+height/2, insets.left+width, insets.top+height/2); if (!ctrl) return; g.drawRect(min(mpX, mdX), min(mpY, mdY), abs(mpX - mdX), abs(mpY - mdY)); }
private void newImage() { long milli = printPoint(); image = new BufferedImage(width, height, TYPE_INT_RGB); int maxX = minX + width; int maxY = minY + height; for (int x = minX; x < maxX; x+=1) { double r = x * Zoom; for (int y = minY; y < maxY; y+=1) { double i = y * Zoom; //System.out.printf("%+f%+fi\n", r, i); // 0f 1/6f 1/3f 1/2f 2/3f 5/6f //straight -> red yellow green cian blue magenta <- reverse image.setRGB(x-minX, y-minY, color(r, i, 360, false, 2/3f)); } } newImage = false; done(milli); }
private long printPoint() { return printPoint(Zoom, minX, minY); } private long printPoint(Point point) { return printPoint(point.Zoom, point.minX, point.minY); } private long printPoint(double Zoom, int minX, int minY) { return printPoint(Zoom, minX*Zoom, minY*Zoom); } private long printPoint(Object ... point) { System.out.printf(ROOT, "{%.16e, %.16g, %.16g},", point); return currentTimeMillis(); }
private void done(long milli) { milli = currentTimeMillis() - milli; System.out.println(" // " + milli + "ms done!"); }
private int color(double r0, double i0, int max, boolean straight, float shift) { int n = -1; double r=0, i=0, r2=0, i2=0; do { i = r*(i+i) + i0; r = r2-i2 + r0; r2 = r*r; i2 = i*i; } while (++n < max && r2 + i2 < 4); return n == max ? black.getRGB() : HSBtoRGB(shift + (float) (straight ? n : max-n) / max * 11/12f + (straight ? 0f : 1/12f), 1, 1) ; } }</lang>
JavaScript
This needs the canvas tag of HTML 5 (it will not run on IE8 and lower or old browsers).
The code can be run directly from the Javascript console in modern browsers by copying and pasting it.
<lang javascript>function mandelIter(cx, cy, maxIter) {
var x = 0.0; var y = 0.0; var xx = 0; var yy = 0; var xy = 0;
var i = maxIter;
while (i-- && xx + yy <= 4) {
xy = x * y;
xx = x * x;
yy = y * y;
x = xx - yy + cx;
y = xy + xy + cy;
}
return maxIter - i;
}
function mandelbrot(canvas, xmin, xmax, ymin, ymax, iterations) {
var width = canvas.width; var height = canvas.height;
var ctx = canvas.getContext('2d');
var img = ctx.getImageData(0, 0, width, height);
var pix = img.data;
for (var ix = 0; ix < width; ++ix) {
for (var iy = 0; iy < height; ++iy) {
var x = xmin + (xmax - xmin) * ix / (width - 1);
var y = ymin + (ymax - ymin) * iy / (height - 1);
var i = mandelIter(x, y, iterations);
var ppos = 4 * (width * iy + ix);
if (i > iterations) {
pix[ppos] = 0;
pix[ppos + 1] = 0;
pix[ppos + 2] = 0;
} else {
var c = 3 * Math.log(i) / Math.log(iterations - 1.0);
if (c < 1) {
pix[ppos] = 255 * c;
pix[ppos + 1] = 0;
pix[ppos + 2] = 0;
}
else if ( c < 2 ) {
pix[ppos] = 255;
pix[ppos + 1] = 255 * (c - 1);
pix[ppos + 2] = 0;
} else {
pix[ppos] = 255;
pix[ppos + 1] = 255;
pix[ppos + 2] = 255 * (c - 2);
}
}
pix[ppos + 3] = 255;
}
}
ctx.putImageData(img, 0, 0);
}
var canvas = document.createElement('canvas'); canvas.width = 900; canvas.height = 600;
document.body.insertBefore(canvas, document.body.childNodes[0]);
mandelbrot(canvas, -2, 1, -1, 1, 1000);</lang>
- Output:
with default parameters
ES6/WebAssembly
With ES6 and WebAssembly, the program can run faster. Of course, this requires a compiled WASM file, but one can easily build one for instance with the WebAssembly explorer
<lang javascript>var mandelIter; fetch("./mandelIter.wasm")
.then(res => {
if (res.ok) return res.arrayBuffer();
throw new Error('Unable to fetch WASM.');
})
.then(bytes => { return WebAssembly.compile(bytes); })
.then(module => { return WebAssembly.instantiate(module); })
.then(instance => { WebAssembly.instance = instance; draw(); })
function mandelbrot(canvas, xmin, xmax, ymin, ymax, iterations) {
// ... var i = WebAssembly.instance.exports.mandelIter(x, y, iterations); // ...
}
function draw() {
// canvas initialization if necessary // ... mandelbrot(canvas, -2, 1, -1, 1, 1000); // ...
}</lang>
jq
The Mandelbrot function as defined here is similar to the JavaScript implementation but generates SVG. The resulting picture is the same.
Preliminaries <lang jq># SVG STUFF
def svg(id; width; height):
"<svg width='\(width // "100%")' height='\(height // "100%") '
id='\(id)'
xmlns='http://www.w3.org/2000/svg'>";
def pixel(x;y;r;g;b;a): "<circle cx='\(x)' cy='\(y)' r='1' fill='rgb(\(r|floor),\(g|floor),\(b|floor))' />";
- "UNTIL"
# As soon as "condition" is true, then emit . and stop: def do_until(condition; next): def u: if condition then . else (next|u) end; u;
</lang><lang jq> def Mandeliter( cx; cy; maxiter ):
# [i, x, y, x^2+y^2]
[ maxiter, 0.0, 0.0, 0.0 ]
| do_until( .[0] == 0 or .[3] > 4;
.[1] as $x | .[2] as $y
| ($x * $y) as $xy
| ($x * $x) as $xx
| ($y * $y) as $yy
| [ (.[0] - 1), # i
($xx - $yy + cx), # x
($xy + $xy + cy), # y
($xx+$yy) # xx+yy
] )
| maxiter - .[0];
- width and height should be specified as the number of pixels.
- obj == { xmin: _, xmax: _, ymin: _, ymax: _ }
def Mandelbrot( obj; width; height; iterations ):
def pixies:
range(0; width) as $ix
| (obj.xmin + ((obj.xmax - obj.xmin) * $ix / (width - 1))) as $x
| range(0; height) as $iy
| (obj.ymin + ((obj.ymax - obj.ymin) * $iy / (height - 1))) as $y
| Mandeliter( $x; $y; iterations ) as $i
| if $i == iterations then
pixel($ix; $iy; 0; 0; 0; 255)
else
(3 * ($i|log)/((iterations - 1.0)|log)) as $c # redness
| if $c < 1 then
pixel($ix;$iy; 255*$c; 0; 0; 255)
elif $c < 2 then
pixel($ix;$iy; 255; 255*($c-1); 0; 255)
else
pixel($ix;$iy; 255; 255; 255*($c-2); 255)
end
end;
svg("mandelbrot"; "100%"; "100%"),
pixies,
"</svg>";</lang>
Example: <lang jq> Mandelbrot( {"xmin": -2, "xmax": 1, "ymin": -1, "ymax":1}; 900; 600; 1000 ) </lang>
Execution:
$ jq -n -r -f mandelbrot.jq > mandelbrot.svg
The output can be viewed in a web browser such as Chrome, Firefox, or Safari.
Julia
Generates an ASCII representation: <lang python>function mandelbrot(a)
z = 0
for i=1:50
z = z^2 + a
end
return z
end
for y=1.0:-0.05:-1.0
for x=-2.0:0.0315:0.5
abs(mandelbrot(complex(x, y))) < 2 ? print("*") : print(" ")
end
println()
end</lang>
This generates a PNG image file: <lang python>using Images
@inline function hsv2rgb(h, s, v)
c = v * s
x = c * (1 - abs(((h/60) % 2) - 1))
m = v - c
r,g,b = if h < 60 (c, x, 0)
elseif h < 120 (x, c, 0)
elseif h < 180 (0, c, x)
elseif h < 240 (0, x, c)
elseif h < 300 (x, 0, c)
else (c, 0, x) end
(r + m), (b + m), (g + m)
end
function mandelbrot()
w = 1600
h = 1200
zoom = 0.5
moveX = -0.5
moveY = 0
maxIter = 30
img = Array{RGB{Float64},2}(undef,h,w)
for x in 1:w
for y in 1:h
i = maxIter
z = c = Complex( (2*x - w) / (w * zoom) + moveX,
(2*y - h) / (h * zoom) + moveY )
while abs(z) < 2 && (i -= 1) > 0
z = z^2 + c
end
r,g,b = hsv2rgb(i / maxIter * 360, 1, i / maxIter)
img[y,x] = RGB{Float64}(r, g, b)
end
end
return img
end
img = mandelbrot() save("mandelbrot.png", img)</lang>
Mandelbrot Set with Julia Animation
This is an extension of the corresponding R section: e^(-|z|)-smoothing was added. <lang julia>using Plots gr(aspect_ratio=:equal, legend=false, axis=false, ticks=false, dpi=100)
d, h = 400, 300 # pixel density (= image width) and image height n, r = 40, 1000 # number of iterations and escape radius (r > 2)
x = range(-1.0, 1.0, length=d+1) y = range(-h/d, h/d, length=h+1)
C = 2.0 .* (x' .+ y .* im) .- 0.5 Z, S = zero(C), zeros(size(C))
animation = Animation() smoothing = Animation()
for k in 1:n
M = abs.(Z) .< r S[M] = S[M] .+ exp.(.-abs.(Z[M])) Z[M] = Z[M] .^ 2 .+ C[M] heatmap(exp.(.-abs.(Z)), c=:jet) frame(animation) heatmap(S .+ exp.(.-abs.(Z)), c=:jet) frame(smoothing)
end
gif(animation, "Mandelbrot_animation.gif", fps=2) gif(smoothing, "Mandelbrot_smoothing.gif", fps=2)</lang>
Normalized Iteration Count, Distance Estimation and Mercator Maps
This is just a translation of the corresponding Python section. The e^(-|z|)-smoothing, normalized iteration count and exterior distance estimation algorithms are used. <lang julia>using Plots gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)
d, h = 800, 500 # pixel density (= image width) and image height n, r = 200, 500 # number of iterations and escape radius (r > 2)
x = range(0, 2, length=d+1) y = range(0, 2 * h / d, length=h+1)
A, B = collect(x .- 1), collect(y .- h / d) C = (2.0 .+ 1.0im) .* (A' .+ B .* im) .- 0.5
Z, dZ = zero(C), zero(C) S, T, D = zeros(size(C)), zeros(size(C)), zeros(size(C))
for k in 1:n
M = abs2.(Z) .< abs2(r) S[M], T[M] = S[M] .+ exp.(.- abs.(Z[M])), T[M] .+ 1 Z[M], dZ[M] = Z[M] .^ 2 .+ C[M], 2 .* Z[M] .* dZ[M] .+ 1
end
heatmap(S .^ 0.1, c=:jet) savefig("Mandelbrot_set_1.png")
N = abs.(Z) .> r # normalized iteration count T[N] = T[N] .- log2.(log.(abs.(Z[N])) ./ log(r))
heatmap(T .^ 0.1, c=:jet) savefig("Mandelbrot_set_2.png")
N = abs.(Z) .> 2 # exterior distance estimation D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])
heatmap(D .^ 0.1, c=:jet) savefig("Mandelbrot_set_3.png")</lang>
A small change in the above code creates Mercator maps of the Mandelbrot set. However, the zoom images are of poor quality. GR cannot create them in good quality, and if you choose PyPlot as the backend, you can also run the Python version directly (thanks to NumPy, it's just as fast as Julia). <lang julia>using Plots gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)
d, h = 200, 1200 # pixel density (= image width) and image height n, r = 800, 1000 # number of iterations and escape radius (r > 2)
x = range(0, 2, length=d+1) y = range(0, 2 * h / d, length=h+1)
A, B = collect(x .* pi), collect(y .* pi) C = (.- 8.0im) .* exp.((A' .+ B .* im) .* im) .- 0.7436636774 .+ 0.1318632144im
Z, dZ = zero(C), zero(C) D = zeros(size(C))
for k in 1:n
M = abs2.(Z) .< abs2(r) Z[M], dZ[M] = Z[M] .^ 2 .+ C[M], 2 .* Z[M] .* dZ[M] .+ 1
end
N = abs.(Z) .> 2 # exterior distance estimation D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])
heatmap(D' .^ 0.1, c=:nipy_spectral) savefig("Mercator_Mandelbrot_map.png")
X, Y = real(C), imag(C) # zoom images (adjust circle size 120 and zoom level 20 as needed) R, c, z = 120 .* 2 ./ d .* pi .* exp.(.- B), min(d, h) + 1, max(0, h - d) ÷ 20
gr(aspect_ratio=:equal, axis=false, ticks=false, legend=false, markerstrokewidth=0, dpi=200) p1 = scatter(X[1*z+1:1*z+c,:], Y[1*z+1:1*z+c,:], markersize=R[1:c,:], marker_z=D[1*z+1:1*z+c,:].^0.5, c=:nipy_spectral) p2 = scatter(X[2*z+1:2*z+c,:], Y[2*z+1:2*z+c,:], markersize=R[1:c,:], marker_z=D[2*z+1:2*z+c,:].^0.4, c=:nipy_spectral) p3 = scatter(X[3*z+1:3*z+c,:], Y[3*z+1:3*z+c,:], markersize=R[1:c,:], marker_z=D[3*z+1:3*z+c,:].^0.3, c=:nipy_spectral) p4 = scatter(X[4*z+1:4*z+c,:], Y[4*z+1:4*z+c,:], markersize=R[1:c,:], marker_z=D[4*z+1:4*z+c,:].^0.2, c=:nipy_spectral) plot(p1, p2, p3, p4, layout=(2, 2)) savefig("Mercator_Mandelbrot_plot.png")</lang>
But in Julia, unlike in Python, it's quite easy to extend the zoom range much further than h > 5.5d by using BigFloats. This is currently not possible with NumPy. However, this is at the expense of computing time, so that an optimization of the loop would make sense. See also the album Mercator Mandelbrot Maps by Anders Sandberg . <lang julia>using Plots gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)
setprecision(BigFloat, 128) # set precision to 128 bits (reduced) setrounding(BigFloat, RoundNearest) # set rounding mode (default)
d, h = 200, 2000 # pixel density (= image width) and image height n, r = 100, 1000 # number of iterations and escape radius (r > 2)
x = range(BigFloat(0), BigFloat(2), length=d+1) y = range(BigFloat(0), BigFloat(2) * h / d, length=h+1)
A, B = collect(x .* pi), collect(y .* pi) C = 4.0 .* exp.(.- A .* im)' .* exp.(.- B) .+ 1.0im
Z, dZ = zero(C), zero(C) D = zeros(size(C))
abs2_Z = abs2.(Z) for k in 1:n
M = abs2_Z .< abs2(r) Z[M], dZ[M] = Z[M] .^ 2 .+ C[M], 2 .* Z[M] .* dZ[M] .+ 1 abs2_Z[M] = abs2.(Z[M])
end
N = abs.(Z) .> 2 # exterior distance estimation D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])
heatmap(D' .^ 0.05, c=:nipy_spectral) savefig("Deep_Mercator_Mandelbrot.png")</lang>
The MultiFloats.jl program library can be used to speed up the calculation. To do this, however, the complex exponential function must be broken down into real functions (cos, sin, exp) and these functions must be calculated with BigFloats. However, the one-off calculation with a few BigFloats before the loop only has a minor impact on the calculation speed. Since the number pi is missing from the MultiFloats.jl library, 2*pi is replaced by 4*(pi/2) = 4*acos(0). See also the picture Deeper Mercator Mandelbrot by Anders Sandberg. <lang julia>using Plots gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)
using MultiFloats MultiFloats.use_bigfloat_transcendentals()
d, h = 60, 600 # pixel density (= image width) and image height n, r = 8000, 10000 # number of iterations and escape radius (r > 2)
a = Float64x2("-1.256827152259138864846434197797294538253477389787308085590211144291") b = Float64x2(".37933802890364143684096784819544060002129071484943239316486643285025")
x = range(zero(a), acos(zero(a)), length=d+1) y = range(zero(b), acos(zero(b)) * h / d, length=h+1)
A, B = collect(x .* 4), collect(y .* 4) C = 4.0 .* (cos.(A) .- sin.(A) .* im)' .* exp.(.- B) .+ a .+ b .* im
Z, dZ = zero(C), zero(C) D = zeros(size(C))
abs2_Z = abs2.(Z) for k in 1:n
M = abs2_Z .< abs2(r) Z[M], dZ[M] = Z[M] .^ 2 .+ C[M], 2 .* Z[M] .* dZ[M] .+ 1 abs2_Z[M] = abs2.(Z[M])
end
N = abs.(Z) .> 2 # exterior distance estimation D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])
heatmap(D' .^ 0.05, c=:nipy_spectral) savefig("Deeper_Mercator_Mandelbrot.png")</lang>
Kotlin
<lang scala>// version 1.1.2
import java.awt.Graphics import java.awt.image.BufferedImage import javax.swing.JFrame
class Mandelbrot: JFrame("Mandelbrot Set") {
companion object {
private const val MAX_ITER = 570
private const val ZOOM = 150.0
}
private val img: BufferedImage
init {
setBounds(100, 100, 800, 600)
isResizable = false
defaultCloseOperation = EXIT_ON_CLOSE
img = BufferedImage(width, height, BufferedImage.TYPE_INT_RGB)
for (y in 0 until height) {
for (x in 0 until width) {
var zx = 0.0
var zy = 0.0
val cX = (x - 400) / ZOOM
val cY = (y - 300) / ZOOM
var iter = MAX_ITER
while (zx * zx + zy * zy < 4.0 && iter > 0) {
val tmp = zx * zx - zy * zy + cX
zy = 2.0 * zx * zy + cY
zx = tmp
iter--
}
img.setRGB(x, y, iter or (iter shl 7))
}
}
}
override fun paint(g: Graphics) {
g.drawImage(img, 0, 0, this)
}
}
fun main(args: Array<String>) {
Mandelbrot().isVisible = true
}</lang>
LabVIEW
Lang5
<lang Lang5>
- d2c(*,*) 2 compress 'c dress ; # Make a complex number.
- iterate(c) [0 0](c) "dup * over +" steps reshape execute ;
- print_line(*) "#*+-. " "" split swap subscript "" join . "\n" . ;
75 iota 45 - 20 / # x coordinates 29 iota 14 - 10 / # y cordinates 'd2c outer # Make complex matrix.
10 'steps set # How many iterations?
iterate abs int 5 min 'print_line apply # Compute & print </lang>
Lambdatalk
Lambdatalk working in any web browser has access to javascript and could draw the mandelbrot set very quickly in a canvas, for instance here: - http: // lambdaway.free.fr/lambdawalks/?view=mandel_canvas
Here we show a pure lambdatalk code, a slow but minimalistic and easy to understand version without the burden of any canvas. We just compute if a point is inside or outside the mandelbrot set and just write "o" or "." directly in the wiki page.
<lang Scheme> {def mandel
{def mandel.r
{lambda {:iter :cx :cy :norm :x :y :count}
{if {> :count :iter} // then norm < 4
then o // inside the set
else {if {> :norm 4} // then iter > max
then . // outside the set
else {let { {:cx :cx} {:cy :cy} {:iter :iter}
{:X {+ {* :x :x} -{* :y :y} :cx}} // compute
{:Y {+ {* 2 :x :y} :cy}} // z = z^2+c
{:count {+ :count 1}}
} {mandel.r :iter :cx :cy
{+ {* :X :X} {* :Y :Y}} // the norm
:X :Y :count} }}}}}
{lambda {:iter :cx :cy}
{mandel.r :iter
{+ {* :cx 0.05} -1.50} // centering the set
{+ {* :cy 0.05} -0.75} // inside the frame
0 0 0 0} }}
-> mandel
We call mandel directly in the wiki page
{S.map {lambda {:i} {br} // loop on y
{S.map {{lambda {:i :j} // loop on x
{mandel 20 :i :j}} :i} // compute
{S.serie 0 30}}} // x resolution
{S.serie 0 40}} // y resolution
</lang>
. . . . . . . . . . . . . . . o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o o o . . . . . . . . . . . . . . . . . . . . . . . . . . . . o o o . . . . . . . . . . . . . . . . . . . . . . . . . . . o o o o o . . . . . . . . . . . . . . . . . . . . . . . . . o o o o o o o . . . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o . . . . . . . . . . . . . . . . . . . . . o o o o o o o o o . . . . . . . . . . . . . . . . . . . . . . . o o o o o o o . . . . . . . . . . . . . . . . . . . . . . . . o o o o o o o . . . . . . . . . . . . . . . . . . o . o o o o o o o o o o o o o o o . o . . . . . . . . . . . . o . o o o o o o o o o o o o o o o . o . . . . . . . . . . . . o o o o o o o o o o o o o o o o o o o . . . . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . . . o o o o o o o o o o o o o o o o o o o o o o o . . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o o . . o . o o o o o o o o o o o o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o o o o o o o o o o o o . o . . o o o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . o . o o o o o o o o o o o o o o o o o o o o o o o . o . . . . . . o o o o o o o o o o o o o o o o o o o o o o o . . . . . . . . o o o o o o o o o o o o o o o o o o o o o o o . . . . . . . . o o o o o o o o o o o . o o o o o o o o o o o . . . . . . . . o . . . o o o o o o . . . o o o o o o . . . o . . . . . . . . . . . . o . . o o . . . . . o o . . o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lasso
<lang Lasso> define mandelbrotBailout => 16 define mandelbrotMaxIterations => 1000
define mandelbrotIterate(x, y) => { local(cr = #y - 0.5, ci = #x, zi = 0.0, zr = 0.0, i = 0, temp, zr2, zi2)
{ ++#i; #temp = #zr * #zi #zr2 = #zr * #zr #zi2 = #zi * #zi
#zi2 + #zr2 > mandelbrotBailout? return #i #i > mandelbrotMaxIterations? return 0
#zr = #zr2 - #zi2 + #cr #zi = #temp + #temp + #ci
currentCapture->restart }() }
define mandelbrotTest() => { local(x, y = -39.0) { stdout('\n') #x = -39.0 { mandelbrotIterate(#x / 40.0, #y / 40.0) == 0? stdout('*') | stdout(' '); ++#x #x <= 39.0? currentCapture->restart }(); ++#y
#y <= 39.0? currentCapture->restart }() stdout('\n') }
mandelbrotTest </lang>
- Output:
*
*
*
*
*
***
*****
*****
***
*
*********
*************
***************
*********************
*********************
*******************
*******************
*******************
*******************
***********************
*******************
*******************
*********************
*******************
*******************
*****************
***************
*************
*********
*
***************
***********************
* ************************* *
*****************************
* ******************************* *
*********************************
***********************************
***************************************
*** ***************************************** ***
*************************************************
***********************************************
*********************************************
*********************************************
***********************************************
***********************************************
***************************************************
*************************************************
*************************************************
***************************************************
***************************************************
* *************************************************** *
***** *************************************************** *****
****** *************************************************** ******
******* *************************************************** *******
***********************************************************************
********* *************************************************** *********
****** *************************************************** ******
***** *************************************************** *****
***************************************************
***************************************************
***************************************************
***************************************************
*************************************************
*************************************************
***************************************************
***********************************************
***********************************************
*******************************************
*****************************************
*********************************************
**** ****************** ****************** ****
*** **************** **************** ***
* ************** ************** *
*********** ***********
** ***** ***** **
* * * *
LIL
From the source distribution. Produces a PBM, not shown here. <lang tcl>#
- A mandelbrot generator that outputs a PBM file. This can be used to measure
- performance differences between LIL versions and measure performance
- bottlenecks (although keep in mind that LIL is not supposed to be a fast
- language, but a small one which depends on C for the slow parts - in a real
- program where for some reason mandelbrots are required, the code below would
- be written in C). The code is based on the mandelbrot test for the Computer
- Language Benchmarks Game at http://shootout.alioth.debian.org/
- In my current computer (Intel Core2Quad Q9550 @ 2.83GHz) running x86 Linux
- the results are (using the default 256x256 size):
- 2m3.634s - commit 1c41cdf89f4c1e039c9b3520c5229817bc6274d0 (Jan 10 2011)
- To test call
- time ./lil mandelbrot.lil > mandelbrot.pbm
- with an optimized version of lil (compiled with CFLAGS=-O3 make).
set width [expr $argv] if not $width { set width 256 } set height $width set bit_num 0 set byte_acc 0 set iter 50 set limit 2.0
write "P4\n${width} ${height}\n"
for {set y 0} {$y < $height} {inc y} {
for {set x 0} {$x < $width} {inc x} {
set Zr 0.0 Zi 0.0 Tr 0.0 Ti 0.0
set Cr [expr 2.0 * $x / $width - 1.5]
set Ci [expr 2.0 * $y / $height - 1.0]
for {set i 0} {$i < $iter && $Tr + $Ti <= $limit * $limit} {inc i} {
set Zi [expr 2.0 * $Zr * $Zi + $Ci]
set Zr [expr $Tr - $Ti + $Cr]
set Tr [expr $Zr * $Zr]
set Ti [expr $Zi * $Zi]
}
set byte_acc [expr $byte_acc << 1]
if [expr $Tr + $Ti <= $limit * $limit] {
set byte_acc [expr $byte_acc | 1]
}
inc bit_num
if [expr $bit_num == 8] {
writechar $byte_acc
set byte_acc 0
set bit_num 0
} {if [expr $x == $width - 1] {
set byte_acc [expr 8 - $width % 8]
writechar $byte_acc
set byte_acc 0
set bit_num 0
}}
}
}</lang>
Logo
<lang logo>to mandelbrot :left :bottom :side :size
cs setpensize [1 1]
make "inc :side/:size
make "zr :left
repeat :size [
make "zr :zr + :inc
make "zi :bottom
pu
setxy repcount - :size/2 minus :size/2
pd
repeat :size [
make "zi :zi + :inc
setpencolor count.color calc :zr :zi
fd 1 ] ]
end
to count.color :count
;op (list :count :count :count) if :count > 256 [op 0] ; black if :count > 128 [op 7] ; white if :count > 64 [op 5] ; magenta if :count > 32 [op 6] ; yellow if :count > 16 [op 4] ; red if :count > 8 [op 2] ; green if :count > 4 [op 1] ; blue op 3 ; cyan
end
to calc :zr :zi [:count 0] [:az 0] [:bz 0]
if :az*:az + :bz*:bz > 4 [op :count] if :count > 256 [op :count] op (calc :zr :zi (:count + 1) (:zr + :az*:az - :bz*:bz) (:zi + 2*:az*:bz))
end
mandelbrot -2 -1.25 2.5 400</lang>
Lua
Graphical
Needs LÖVE 2D Engine
Zoom in: drag the mouse; zoom out: right click
<lang lua>
local maxIterations = 250
local minX, maxX, minY, maxY = -2.5, 2.5, -2.5, 2.5
local miX, mxX, miY, mxY
function remap( x, t1, t2, s1, s2 )
local f = ( x - t1 ) / ( t2 - t1 ) local g = f * ( s2 - s1 ) + s1 return g;
end function drawMandelbrot()
local pts, a, as, za, b, bs, zb, cnt, clr = {}
for j = 0, hei - 1 do
for i = 0, wid - 1 do
a = remap( i, 0, wid, minX, maxX )
b = remap( j, 0, hei, minY, maxY )
cnt = 0; za = a; zb = b
while( cnt < maxIterations ) do
as = a * a - b * b; bs = 2 * a * b
a = za + as; b = zb + bs
if math.abs( a ) + math.abs( b ) > 16 then break end
cnt = cnt + 1
end
if cnt == maxIterations then clr = 0
else clr = remap( cnt, 0, maxIterations, 0, 255 )
end
pts[1] = { i, j, clr, clr, 0, 255 }
love.graphics.points( pts )
end
end
end function startFractal()
love.graphics.setCanvas( canvas ); love.graphics.clear() love.graphics.setColor( 255, 255, 255 ) drawMandelbrot(); love.graphics.setCanvas()
end function love.load()
wid, hei = love.graphics.getWidth(), love.graphics.getHeight() canvas = love.graphics.newCanvas( wid, hei ) startFractal()
end function love.mousepressed( x, y, button, istouch )
if button == 1 then
startDrag = true; miX = x; miY = y
else
minX = -2.5; maxX = 2.5; minY = minX; maxY = maxX
startFractal()
startDrag = false
end
end function love.mousereleased( x, y, button, istouch )
if startDrag then
local l
if x > miX then mxX = x
else l = x; mxX = miX; miX = l
end
if y > miY then mxY = y
else l = y; mxY = miY; miY = l
end
miX = remap( miX, 0, wid, minX, maxX )
mxX = remap( mxX, 0, wid, minX, maxX )
miY = remap( miY, 0, hei, minY, maxY )
mxY = remap( mxY, 0, hei, minY, maxY )
minX = miX; maxX = mxX; minY = miY; maxY = mxY
startFractal()
end
end function love.draw()
love.graphics.draw( canvas )
end </lang>
ASCII
<lang lua>-- Mandelbrot set in Lua 6/15/2020 db local charmap = { [0]=" ", ".", ":", "-", "=", "+", "*", "#", "%", "@" } for y = -1.3, 1.3, 0.1 do
for x = -2.1, 1.1, 0.04 do
local zi, zr, i = 0, 0, 0
while i < 100 do
if (zi*zi+zr*zr >= 4) then break end
zr, zi, i = zr*zr-zi*zi+x, 2*zr*zi+y, i+1
end
io.write(charmap[i%10])
end
print()
end</lang>
- Output:
...............::::::::::::::::::---------------:::::::::::::::::::::::::::::::: .............:::::::::::---------------------------------::::::::::::::::::::::: ...........::::::::---------------------======+#@#++=====-----:::::::::::::::::: ..........::::::--------------------=======+++**@+@.@*+=====-----::::::::::::::: ........:::::--------------------========++++*#@@#=:%#*++=====------:::::::::::: .......:::--------------------========+++***#%: .:#*+++++===-------::::::::: ......:::------------------=======+++#%@%%%@@ :# %.@@%#***%*+==------:::::::: .....::-----------------====++++++**# = =% %-.*+=#%+==-------:::::: .....:--------------==+++++++++****%:.*+ *%*++==-------::::: ....:--------=====+*#=%##########%%.** .@ +===-------:::: ....---========++++*#@:=*:-@+ .@@ := @ #*===--------::: ...--=======+++++*##%.=: @+. @##+====--------:: ...=======****##% .-=% *#*+====--------:: ...@.:: :: % :@#*++====--------:: ...=======****##% .-=% *#*+====--------:: ...--=======+++++*##%.=: @+. @##+====--------:: ....---========++++*#@:=*:-@+ .@@ := @ #*===--------::: ....:--------=====+*#=%##########%%.** .@ +===-------:::: .....:--------------==+++++++++****%:.*+ *%*++==-------::::: .....::-----------------====++++++**# = =% %-.*+=#%+==-------:::::: ......:::------------------=======+++#%@%%%@@ :# %.@@%#***%*+==------:::::::: .......:::--------------------========+++***#%: .:#*+++++===-------::::::::: ........:::::--------------------========++++*#@@#=:%#*++=====------:::::::::::: ..........::::::--------------------=======+++**@+@.@*+=====-----::::::::::::::: ...........::::::::---------------------======+#@#++=====-----:::::::::::::::::: .............:::::::::::---------------------------------::::::::::::::::::::::: ...............::::::::::::::::::---------------::::::::::::::::::::::::::::::::
M2000 Interpreter
Console is a bitmap so we can plot on it. A subroutine plot different size of pixels so we get Mandelbrot image at final same size for 32X26 for a big pixel of 16x16 pixels to 512x416 for a 1:1 pixel. Iterations for each pixel set to 25. Module can get left top corner as twips, and the size factor from 1 to 16 (size of output is 512x416 pixels for any factor).
<lang M2000 Interpreter> Module Mandelbrot(x=0&,y=0&,z=1&) {
If z<1 then z=1
If z>16 then z=16
Const iXmax=32*z
Const iYmax=26*z
Def single Cx, Cy, CxMin=-2.05, CxMax=0.85, CyMin=-1.2, CyMax=1.2
Const PixelWidth=(CxMax-CxMin)/iXmax, iXm=(iXmax-1)*PixelWidth
Const PixelHeight=(CyMax-CyMin)/iYmax,Ph2=PixelHeight/2
Const Iteration=25
Const EscRadious=2.5, ER2=EscRadious**2
Def single preview
preview=iXmax*twipsX*(z/16)
Def long yp, xp, dx, dy, dx1, dy1
Let dx=twipsx*(16/z), dx1=dx-1
Let dy=twipsy*(16/z), dy1=dy-1
yp=y
For iY=0 to (iYmax-1)*PixelHeight step PixelHeight {
Cy=CyMin+iY
xp=x
if abs(Cy)<Ph2 Then Cy=0
For iX=0 to iXm Step PixelWidth {
Let Cx=CxMin+iX,Zx=0,Zy=0,Zx2=Zx**2,Zy2=Zy**2
For It=Iteration to 1 {Let Zy=2*Zx*Zy+Cy,Zx=Zx2-Zy2+Cx,Zx2=Zx**2,Zy2=Zy**2 :if Zx2+Zy2>ER2 Then exit
}
if it>13 then {it-=13} else.if it=0 then SetPixel(xp,yp,0): xp+=dx : continue
it*=10:SetPixel(xp,yp,color(it, it,255)) :xp+=dx
} : yp+=dy
}
Sub SetPixel()
move number, number: fill dx1, dy1, number
End Sub
} Cls 1,0 sz=(1,2,4,8,16) i=each(sz) While i {
Mandelbrot 250*twipsx,100*twipsy, array(i)
}
</lang>
Version 2 without Subroutine. Also there is a screen refresh every 2 seconds. <lang M2000 Interpreter> Module Mandelbrot(x=0&,y=0&,z=1&) {
If z<1 then z=1
If z>16 then z=16
Const iXmax=32*z
Const iYmax=26*z
Def single Cx, Cy, CxMin=-2.05, CxMax=0.85, CyMin=-1.2, CyMax=1.2
Const PixelWidth=(CxMax-CxMin)/iXmax, iXm=(iXmax-1)*PixelWidth
Const PixelHeight=(CyMax-CyMin)/iYmax,Ph2=PixelHeight/2
Const Iteration=25
Const EscRadious=2.5, ER2=EscRadious**2
Def single preview
preview=iXmax*twipsX*(z/16)
Def long yp, xp, dx, dy, dx1, dy1
Let dx=twipsx*(16/z), dx1=dx-1
Let dy=twipsy*(16/z), dy1=dy-1
yp=y
Refresh 2000
For iY=0 to (iYmax-1)*PixelHeight step PixelHeight {
Cy=CyMin+iY
xp=x
if abs(Cy)<Ph2 Then Cy=0
move xp, yp
For iX=0 to iXm Step PixelWidth {
Let Cx=CxMin+iX,Zx=0,Zy=0,Zx2=Zx**2,Zy2=Zy**2
For It=Iteration to 1 {Let Zy=2*Zx*Zy+Cy,Zx=Zx2-Zy2+Cx,Zx2=Zx**2,Zy2=Zy**2 :if Zx2+Zy2>ER2 Then exit
}
if it>13 then {it-=13} else.if it=0 then fill dx1, dy1, 0: Step 0,-dy1: continue
it*=10:fill dx1, dy1, color(it, it,255): Step 0,-dy1
} : yp+=dy
}
} Cls 1,0 sz=(1,2,4,8,16) i=each(sz) While i {
Mandelbrot 250*twipsx,100*twipsy, array(i)
} </lang>
Maple
<lang>ImageTools:-Embed(Fractals[EscapeTime]:-Mandelbrot(500, -2.0-1.35*I, .7+1.35*I, output = layer1));</lang>
Mathematica / Wolfram Language
The implementation could be better. But this is a start... <lang mathematica>eTime[c_, maxIter_Integer: 100] := Length@NestWhileList[#^2 + c &, 0, Abs@# <= 2 &, 1, maxIter] - 1
DistributeDefinitions[eTime]; mesh = ParallelTable[eTime[x + I*y, 1000], {y, 1.2, -1.2, -0.01}, {x, -1.72, 1, 0.01}]; ReliefPlot[mesh, Frame -> False]</lang> Faster version: <lang mathematica>cf = With[{
mandel = Block[{z = #, c = #},
Catch@Do[If[Abs[z] > 2, Throw@i]; z = z^2 + c, {i, 100}]] &
},
Compile[{},Table[mandel[y + x I], {x, -1, 1, 0.005}, {y, -2, 0.5, 0.005}]]
];
ArrayPlot[cf[]] </lang> Built-in function: <lang mathematica>MandelbrotSetPlot[]</lang>
Mathmap
filter mandelbrot (gradient coloration)
c=ri:(xy/xy:[X,X]*1.5-xy:[0.5,0]);
z=ri:[0,0]; # initial value z0 = 0
# iteration of z
iter=0;
while abs(z)<2 && iter<31
do
z=z*z+c; # z(n+1) = fc(zn)
iter=iter+1
end;
coloration(iter/32) # color of pixel
end
MATLAB
This solution uses the escape time algorithm to determine the coloring of the coordinates on the complex plane. The code can be reduced to a single line via vectorization after the Escape Time Algorithm function definition, but the code becomes unnecessarily obfuscated. Also, this code uses a lot of memory. You will need a computer with a lot of memory to compute the set with high resolution.
<lang MATLAB>function [theSet,realAxis,imaginaryAxis] = mandelbrotSet(start,gridSpacing,last,maxIteration)
%Define the escape time algorithm
function escapeTime = escapeTimeAlgorithm(z0)
escapeTime = 0;
z = 0;
while( (abs(z)<=2) && (escapeTime < maxIteration) )
z = (z + z0)^2;
escapeTime = escapeTime + 1;
end
end
%Define the imaginary axis
imaginaryAxis = (imag(start):imag(gridSpacing):imag(last));
%Define the real axis
realAxis = (real(start):real(gridSpacing):real(last));
%Construct the complex plane from the real and imaginary axes
complexPlane = meshgrid(realAxis,imaginaryAxis) + meshgrid(imaginaryAxis(end:-1:1),realAxis)'.*i;
%Apply the escape time algorithm to each point in the complex plane
theSet = arrayfun(@escapeTimeAlgorithm, complexPlane);
%Draw the set pcolor(realAxis,imaginaryAxis,theSet); shading flat;
end</lang>
To use this function you must specify the:
- lower left hand corner of the complex plane from which to start the image,
- the grid spacing in both the imaginary and real directions,
- the upper right hand corner of the complex plane at which to end the image and
- the maximum iterations for the escape time algorithm.
For example:
- Lower Left Corner: -2.05-1.2i
- Grid Spacing: 0.004+0.0004i
- Upper Right Corner: 0.45+1.2i
- Maximum Iterations: 500
Sample usage: <lang MATLAB>mandelbrotSet(-2.05-1.2i,0.004+0.0004i,0.45+1.2i,500);</lang>
Metapost
<lang Metapost>prologues:=3; outputtemplate:="%j-%c.svg"; outputformat:="svg";
def mandelbrot(expr maxX, maxY) =
max_iteration := 500;
color col[]; for i := 0 upto max_iteration: t := i / max_iteration; col[i] = (t,t,t); endfor;
for px := 0 upto maxX:
for py := 0 upto maxY:
xz := px * 3.5 / maxX - 2.5; % (-2.5,1)
yz := py * 2 / maxY - 1; % (-1,1)
x := 0;
y := 0;
iteration := 0;
forever: exitunless ((x*x + y*y < 4) and (iteration < max_iteration));
xtemp := x*x - y*y + xz;
y := 2*x*y + yz;
x := xtemp;
iteration := iteration + 1;
endfor;
draw (px,py) withpen pencircle withcolor col[iteration];
endfor; endfor;
enddef;
beginfig(1);
mandelbrot(200, 150);
endfig;
end</lang>
Sample usage: <lang Bash>mpost -numbersystem="double" mandelbrot.mp</lang>
MiniScript
<lang MiniScript>ZOOM = 100 MAX_ITER = 40 gfx.clear color.black for y in range(0,200) for x in range(0,300) zx = 0 zy = 0 cx = (x - 200) / ZOOM cy = (y - 100) / ZOOM for iter in range(MAX_ITER) if zx*zx + zy*zy > 4 then break tmp = zx * zx - zy * zy + cx zy = 2 * zx * zy + cy zx = tmp end for if iter then gfx.setPixel x, y, rgb(255-iter*6, 0, iter*6) end if end for end for</lang>
(Will upload an output image as soon as image uploading is fixed.)
Modula-3
<lang modula3>MODULE Mandelbrot EXPORTS Main;
IMPORT Wr, Stdio, Fmt, Word;
CONST m = 50;
limit2 = 4.0;
TYPE UByte = BITS 8 FOR [0..16_FF];
VAR width := 200;
height := 200; bitnum: CARDINAL := 0; byteacc: UByte := 0; isOverLimit: BOOLEAN; Zr, Zi, Cr, Ci, Tr, Ti: REAL;
BEGIN
Wr.PutText(Stdio.stdout, "P4\n" & Fmt.Int(width) & " " & Fmt.Int(height) & "\n");
FOR y := 0 TO height - 1 DO
FOR x := 0 TO width - 1 DO
Zr := 0.0; Zi := 0.0;
Cr := 2.0 * FLOAT(x) / FLOAT(width) - 1.5;
Ci := 2.0 * FLOAT(y) / FLOAT(height) - 1.0;
FOR i := 1 TO m + 1 DO
Tr := Zr*Zr - Zi*Zi + Cr;
Ti := 2.0*Zr*Zi + Ci;
Zr := Tr; Zi := Ti;
isOverLimit := Zr*Zr + Zi*Zi > limit2;
IF isOverLimit THEN EXIT; END;
END;
IF isOverLimit THEN
byteacc := Word.Xor(Word.LeftShift(byteacc, 1), 16_00);
ELSE
byteacc := Word.Xor(Word.LeftShift(byteacc, 1), 16_01);
END;
INC(bitnum);
IF bitnum = 8 THEN
Wr.PutChar(Stdio.stdout, VAL(byteacc, CHAR));
byteacc := 0;
bitnum := 0;
ELSIF x = width - 1 THEN
byteacc := Word.LeftShift(byteacc, 8 - (width MOD 8));
Wr.PutChar(Stdio.stdout, VAL(byteacc, CHAR));
byteacc := 0;
bitnum := 0
END;
Wr.Flush(Stdio.stdout);
END;
END;
END Mandelbrot.</lang>
MySQL
See http://arbitraryscrawl.blogspot.co.uk/2012/06/fractsql.html for an explanation.
<lang mysql> -- Table to contain all the data points CREATE TABLE points (
c_re DOUBLE, c_im DOUBLE, z_re DOUBLE DEFAULT 0, z_im DOUBLE DEFAULT 0, znew_re DOUBLE DEFAULT 0, znew_im DOUBLE DEFAULT 0, steps INT DEFAULT 0, active CHAR DEFAULT 1
);
DELIMITER |
-- Iterate over all the points in the table 'points' CREATE PROCEDURE itrt (IN n INT) BEGIN
label: LOOP
UPDATE points
SET
znew_re=POWER(z_re,2)-POWER(z_im,2)+c_re,
znew_im=2*z_re*z_im+c_im,
steps=steps+1
WHERE active=1;
UPDATE points SET
z_re=znew_re,
z_im=znew_im,
active=IF(POWER(z_re,2)+POWER(z_im,2)>4,0,1)
WHERE active=1;
SET n = n - 1;
IF n > 0 THEN
ITERATE label;
END IF;
LEAVE label;
END LOOP label;
END|
-- Populate the table 'points' CREATE PROCEDURE populate (
r_min DOUBLE, r_max DOUBLE, r_step DOUBLE, i_min DOUBLE, i_max DOUBLE, i_step DOUBLE)
BEGIN
DELETE FROM points;
SET @rl = r_min;
SET @a = 0;
rloop: LOOP
SET @im = i_min;
SET @b = 0;
iloop: LOOP
INSERT INTO points (c_re, c_im)
VALUES (@rl, @im);
SET @b=@b+1;
SET @im=i_min + @b * i_step;
IF @im < i_max THEN
ITERATE iloop;
END IF;
LEAVE iloop;
END LOOP iloop;
SET @a=@a+1;
SET @rl=r_min + @a * r_step;
IF @rl < r_max THEN
ITERATE rloop;
END IF;
LEAVE rloop;
END LOOP rloop;
END|
DELIMITER ;
-- Choose size and resolution of graph -- R_min, R_max, R_step, I_min, I_max, I_step CALL populate( -2.5, 1.5, 0.005, -2, 2, 0.005 );
-- Calculate 50 iterations CALL itrt( 50 );
-- Create the image (/tmp/image.ppm) -- Note, MySQL will not over-write an existing file and you may need -- administrator access to delete or move it SELECT @xmax:=COUNT(c_re) INTO @xmax FROM points GROUP BY c_im LIMIT 1; SELECT @ymax:=COUNT(c_im) INTO @ymax FROM points GROUP BY c_re LIMIT 1; SET group_concat_max_len=11*@xmax*@ymax; SELECT
'P3', @xmax, @ymax, 200,
GROUP_CONCAT(
CONCAT(
IF( active=1, 0, 55+MOD(steps, 200) ), ' ',
IF( active=1, 0, 55+MOD(POWER(steps,3), 200) ), ' ',
IF( active=1, 0, 55+MOD(POWER(steps,2), 200) ) )
ORDER BY c_im ASC, c_re ASC SEPARATOR ' ' )
INTO OUTFILE '/tmp/image.ppm'
FROM points;
</lang>
Nim
Textual version
<lang nim>import complex
proc inMandelbrotSet(c: Complex, maxEscapeIterations = 50): bool =
result = true; var z: Complex for i in 0..maxEscapeIterations: z = z * z + c if abs2(z) > 4: return false
iterator steps(start, step: float, numPixels: int): float =
for i in 0..numPixels: yield start + i.float * step
proc mandelbrotImage(yStart, yStep, xStart, xStep: float, height, width: int): string =
for y in steps(yStart, yStep, height):
for x in steps(xStart, xStep, width):
result.add(if complex(x, y).inMandelbrotSet: '*'
else: ' ')
result.add('\n')
echo mandelbrotImage(1.0, -0.05, -2.0, 0.0315, 40, 80)</lang>
- Output:
**
******
********
******
******** ** *
*** *****************
************************ ***
****************************
******************************
******************************
************************************
* **********************************
** ***** * **********************************
*********** ************************************
************** ************************************
***************************************************
*****************************************************
************************************************************************
*****************************************************
***************************************************
************** ************************************
*********** ************************************
** ***** * **********************************
* **********************************
************************************
******************************
******************************
****************************
************************ ***
*** *****************
******** ** *
******
********
******
**
Graphical version
<lang Nim>import math, complex, lenientops import imageman
const
W = 800 H = 600 Zoom = 0.5 MoveX = -0.5 MoveY = 0.0 MaxIter = 30
func hsvToRgb(h, s, v: float): array[3, float] =
let c = v * s
let x = c * (1 - abs(((h / 60) mod 2) - 1))
let m = v - c
let (r, g, b) = if h < 60: (c, x, 0.0)
elif h < 120: (x, c, 0.0)
elif h < 180: (0.0, c, x)
elif h < 240: (0.0, x, c)
elif x < 300: (x, 0.0, c)
else: (c, 0.0, x)
result = [r + m, g + m, b + m]
var img = initImage[ColorRGBF64](W, H)
for x in 1..W:
for y in 1..H:
var i = MaxIter - 1
let c = complex((2 * x - W) / (W * Zoom) + MoveX, (2 * y - H) / (H * Zoom) + MoveY)
var z = c
while abs(z) < 2 and i > 0:
z = z * z + c
dec i
let color = hsvToRgb(i / MaxIter * 360, 1, i / MaxIter)
img[x - 1, y - 1] = ColorRGBF64(color)
img.savePNG("mandelbrot.png", compression = 9)</lang>
OCaml
<lang ocaml>#load "graphics.cma";;
let mandelbrot xMin xMax yMin yMax xPixels yPixels maxIter =
let rec mandelbrotIterator z c n =
if (Complex.norm z) > 2.0 then false else
match n with
| 0 -> true
| n -> let z' = Complex.add (Complex.mul z z) c in
mandelbrotIterator z' c (n-1) in
Graphics.open_graph
(" "^(string_of_int xPixels)^"x"^(string_of_int yPixels));
let dx = (xMax -. xMin) /. (float_of_int xPixels)
and dy = (yMax -. yMin) /. (float_of_int yPixels) in
for xi = 0 to xPixels - 1 do
for yi = 0 to yPixels - 1 do
let c = {Complex.re = xMin +. (dx *. float_of_int xi);
Complex.im = yMin +. (dy *. float_of_int yi)} in
if (mandelbrotIterator Complex.zero c maxIter) then
(Graphics.set_color Graphics.white;
Graphics.plot xi yi )
else
(Graphics.set_color Graphics.black;
Graphics.plot xi yi )
done
done;;
mandelbrot (-1.5) 0.5 (-1.0) 1.0 500 500 200;;</lang>
Octave
This code runs rather slowly and produces coloured Mandelbrot set by accident (output image).
<lang octave>#! /usr/bin/octave -qf global width = 200; global height = 200; maxiter = 100;
z0 = 0; global cmax = 1 + i; global cmin = -2 - i;
function cs = pscale(c)
global cmax; global cmin; global width; global height; persistent px = (real(cmax-cmin))/width; persistent py = (imag(cmax-cmin))/height; cs = real(cmin) + px*real(c) + i*(imag(cmin) + py*imag(c));
endfunction
ms = zeros(width, height); for x = 0:width-1
for y = 0:height-1
z0 = 0;
c = pscale(x+y*i);
for ic = 1:maxiter
z1 = z0^2 + c;
if ( abs(z1) > 2 ) break; endif
z0 = z1;
endfor
ms(x+1, y+1) = ic/maxiter;
endfor
endfor
saveimage("mandel.ppm", round(ms .* 255).', "ppm");</lang>
A bit faster than the above implementation
<lang octave> function z = mandelbrot()
% to view the image call "image(mandelbrot())" width = 500; height = 500; z = zeros(width, height); c = zeros(width, height);
xi = 1;
for x = linspace(-2, 2, width)
yi = 1;
for y = linspace(-2, 2, height)
c(yi, xi) = x+y*i; yi += 1;
end
xi += 1;
end
for iter = 1:50
z = z.*z + c;
end
z = abs(z);
end </lang>
Ol
<lang scheme> (define x-size 59) (define y-size 21) (define min-im -1) (define max-im 1) (define min-re -2) (define max-re 1)
(define step-x (/ (- max-re min-re) x-size)) (define step-y (/ (- max-im min-im) y-size))
(for-each (lambda (y)
(let ((im (+ min-im (* step-y y))))
(for-each (lambda (x)
(let*((re (+ min-re (* step-x x)))
(zr (inexact re))
(zi (inexact im)))
(let loop ((n 0) (zi zi) (zr zr))
(let ((a (* zr zr))
(b (* zi zi)))
(cond
((> (+ a b) 4)
(display (string (- 62 n))))
((= n 30)
(display (string (- 62 n))))
(else
(loop (+ n 1) (+ (* 2 zr zi) im) (- (+ a re) b))))))))
(iota x-size))
(print)))
(iota y-size))
</lang>
Output:
>>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<========== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======= >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= ><<<<;;;;;:::972456-567763 +9;;<<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< ><<<<;;;;;:::972456-567763 +9;;<<<<<<< >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<=======
OpenEdge/Progress
<lang Progress (OpenEdge ABL)>DEFINE VARIABLE print_str AS CHAR NO-UNDO INIT . DEFINE VARIABLE X1 AS DECIMAL NO-UNDO INIT 50. DEFINE VARIABLE Y1 AS DECIMAL NO-UNDO INIT 21. DEFINE VARIABLE X AS DECIMAL NO-UNDO. DEFINE VARIABLE Y AS DECIMAL NO-UNDO. DEFINE VARIABLE N AS DECIMAL NO-UNDO. DEFINE VARIABLE I3 AS DECIMAL NO-UNDO. DEFINE VARIABLE R3 AS DECIMAL NO-UNDO. DEFINE VARIABLE Z1 AS DECIMAL NO-UNDO. DEFINE VARIABLE Z2 AS DECIMAL NO-UNDO. DEFINE VARIABLE A AS DECIMAL NO-UNDO. DEFINE VARIABLE B AS DECIMAL NO-UNDO. DEFINE VARIABLE I1 AS DECIMAL NO-UNDO INIT -1.0. DEFINE VARIABLE I2 AS DECIMAL NO-UNDO INIT 1.0. DEFINE VARIABLE R1 AS DECIMAL NO-UNDO INIT -2.0. DEFINE VARIABLE R2 AS DECIMAL NO-UNDO INIT 1.0. DEFINE VARIABLE S1 AS DECIMAL NO-UNDO. DEFINE VARIABLE S2 AS DECIMAL NO-UNDO.
S1 = (R2 - R1) / X1.
S2 = (I2 - I1) / Y1.
DO Y = 0 TO Y1 - 1:
I3 = I1 + S2 * Y.
DO X = 0 TO X1 - 1:
R3 = R1 + S1 * X.
Z1 = R3.
Z2 = I3.
DO N = 0 TO 29:
A = Z1 * Z1.
B = Z2 * Z2.
IF A + B > 4.0 THEN
LEAVE.
Z2 = 2 * Z1 * Z2 + I3.
Z1 = A - B + R3.
END.
print_str = print_str + CHR(62 - N).
END.
print_str = print_str + '~n'.
END.
OUTPUT TO "C:\Temp\out.txt".
MESSAGE print_str.
OUTPUT CLOSE.
</lang>Example :
PARI/GP
Define function mandelbrot(): <lang parigp>mandelbrot() = {
forstep(y=-1, 1, 0.05,
forstep(x=-2, 0.5, 0.0315,
print1(((c)->my(z=c);for(i=1,20,z=z*z+c;if(abs(z)>2,return(" ")));"#")(x+y*I)));
print());
}</lang>
Output:
gp > mandelbrot()
#
# ### #
########
#########
######
## ## ############ #
### ################### #
#############################
############################
################################
################################
#################################### #
# # ###################################
########### ###################################
########### #####################################
############## ####################################
####################################################
######################################################
#########################################################################
######################################################
####################################################
############## ####################################
########### #####################################
########### ###################################
# # ###################################
#################################### #
################################
################################
############################
#############################
### ################### #
## ## ############ #
######
#########
########
# ### #
#
Pascal
<lang pascal>program mandelbrot;
{$IFDEF FPC}
{$MODE DELPHI}
{$ENDIF}
const
ixmax = 800; iymax = 800; cxmin = -2.5; cxmax = 1.5; cymin = -2.0; cymax = 2.0; maxcolorcomponentvalue = 255; maxiteration = 200; escaperadius = 2;
type
colortype = record
red : byte;
green : byte;
blue : byte;
end;
var
ix, iy : integer; cx, cy : real; pixelwidth : real = (cxmax - cxmin) / ixmax; pixelheight : real = (cymax - cymin) / iymax; filename : string = 'new1.ppm'; comment : string = '# '; outfile : textfile; color : colortype; zx, zy : real; zx2, zy2 : real; iteration : integer; er2 : real = (escaperadius * escaperadius);
begin
{$I-}
assign(outfile, filename);
rewrite(outfile);
if ioresult <> 0 then
begin
{$IFDEF FPC}
writeln(stderr, 'Unable to open output file: ', filename);
{$ELSE}
writeln('ERROR: Unable to open output file: ', filename);
{$ENDIF}
exit;
end;
writeln(outfile, 'P6');
writeln(outfile, ' ', comment);
writeln(outfile, ' ', ixmax);
writeln(outfile, ' ', iymax);
writeln(outfile, ' ', maxcolorcomponentvalue);
for iy := 1 to iymax do
begin
cy := cymin + (iy - 1)*pixelheight;
if abs(cy) < pixelheight / 2 then cy := 0.0;
for ix := 1 to ixmax do
begin
cx := cxmin + (ix - 1)*pixelwidth;
zx := 0.0;
zy := 0.0;
zx2 := zx*zx;
zy2 := zy*zy;
iteration := 0;
while (iteration < maxiteration) and (zx2 + zy2 < er2) do
begin
zy := 2*zx*zy + cy;
zx := zx2 - zy2 + cx;
zx2 := zx*zx;
zy2 := zy*zy;
iteration := iteration + 1;
end;
if iteration = maxiteration then
begin
color.red := 0;
color.green := 0;
color.blue := 0;
end
else
begin
color.red := 255;
color.green := 255;
color.blue := 255;
end;
write(outfile, chr(color.red), chr(color.green), chr(color.blue));
end;
end;
close(outfile);
end.</lang>
Perl
translation / optimization of the ruby solution <lang perl>use Math::Complex;
sub mandelbrot {
my ($z, $c) = @_[0,0];
for (1 .. 20) {
$z = $z * $z + $c;
return $_ if abs $z > 2;
}
}
for (my $y = 1; $y >= -1; $y -= 0.05) {
for (my $x = -2; $x <= 0.5; $x += 0.0315)
{print mandelbrot($x + $y * i) ? ' ' : '#'}
print "\n"
}</lang>
Phix
- Ascii
This is included in the distribution (with some extra validation) as demo\mandle.exw <lang Phix>-- -- Mandlebrot set in ascii art demo. -- constant b=" .:,;!/>)|&IH%*#" atom r, i, c, C, z, Z, t, k
for y=30 to 0 by -1 do
C = y*0.1-1.5
puts(1,'\n')
for x=0 to 74 do
c = x*0.04-2
z = 0
Z = 0
r = c
i = C
k = 0
while k<112 do
t = z*z-Z*Z+r
Z = 2*z*Z+i
z = t
if z*z+Z*Z>10 then exit end if
k += 1
end while
puts(1,b[remainder(k,16)+1])
end for
end for</lang>
- Graphical
This is included in the distribution as demo\arwendemo\mandel.exw <lang Phix>include arwen.ew include ..\arwen\dib256.ew
constant HelpText = "Left-click drag with the mouse to move the image.\n"&
" (the image is currently only redrawn on mouseup).\n"&
"Right-click-drag with the mouse to select a region to zoom in to.\n"&
"Use the mousewheel to zoom in and out (nb: can be slow).\n"&
"Press F2 to select iterations, higher==more detail but slower.\n"&
"Resize the window as you please, but note that going fullscreen, \n"&
"especially at high iteration, may mean a quite long draw time.\n"&
"Press Escape to close the window."
procedure Help()
void = messageBox("Mandelbrot Set",HelpText,MB_OK)
end procedure
integer cWidth = 520 -- client area width integer cHeight = 480 -- client area height
constant Main = create(Window, "Mandelbrot Set", 0, 0, 50, 50, cWidth+16, cHeight+38, 0),
mainHwnd = getHwnd(Main),
mainDC = getPrivateDC(Main),
mIter = create(Menu, "", 0, 0, 0,0,0,0,0),
iterHwnd = getHwnd(mIter),
mIter50 = create(MenuItem,"50 (fast, low detail)", 0, mIter, 0,0,0,0,0),
mIter100 = create(MenuItem,"100 (default)", 0, mIter, 0,0,0,0,0),
mIter500 = create(MenuItem,"500", 0, mIter, 0,0,0,0,0),
mIter1000 = create(MenuItem,"1000 (slow, high detail)",0, mIter, 0,0,0,0,0),
m50to1000 = {mIter50,mIter100,mIter500,mIter1000},
i50to1000 = { 50, 100, 500, 1000}
integer mainDib = 0
constant whitePen = c_func(xCreatePen, {0,1,BrightWhite}) constant NULL_BRUSH = 5,
NullBrushID = c_func(xGetStockObject,{NULL_BRUSH})
atom t0 integer iter atom x0, y0 -- top-left coords to draw atom scale -- controls width/zoom
procedure init()
x0 = -2
y0 = -1.25
scale = 2.5/cHeight
iter = 100
void = c_func(xSelectObject,{mainDC,whitePen})
void = c_func(xSelectObject,{mainDC,NullBrushID})
end procedure init()
function in_set(atom x, atom y) atom u,t
if x>-0.75 then
u = x-0.25
t = u*u+y*y
return ((2*t+u)*(2*t+u)>t)
else
return ((x+1)*(x+1)+y*y)>0.0625
end if
end function
function pixel_colour(atom x0, atom y0, integer iter) integer count = 1 atom x = 0, y = 0
while (count<=iter) and (x*x+y*y<4) do
count += 1
{x,y} = {x*x-y*y+x0,2*x*y+y0}
end while
if count<=iter then return count end if
return 0
end function
procedure mandel(atom x0, atom y0, atom scale) atom x,y integer c
t0 = time()
y = y0
for yi=1 to cHeight do
x = x0
for xi=1 to cWidth do
c = 0 -- default to black
if in_set(x,y) then
c = pixel_colour(x,y,iter)
end if
setDibPixel(mainDib, xi, yi, c)
x += scale
end for
y += scale
end for
end procedure
integer firsttime = 1 integer drawBox = 0 integer drawTime = 0
procedure newDib() sequence pal
if mainDib!=0 then
{} = deleteDib(mainDib)
end if
mainDib = createDib(cWidth, cHeight)
pal = repeat({0,0,0},256)
for i=2 to 256 do
pal[i][1] = i*5
pal[i][2] = 0
pal[i][3] = i*10
end for
setDibPalette(mainDib, 1, pal)
mandel(x0,y0,scale)
drawTime = 2
end procedure
procedure reDraw()
setText(Main,"Please Wait...") mandel(x0,y0,scale) drawTime = 2 repaintWindow(Main,False)
end procedure
procedure zoom(integer z)
while z do
if z>0 then
scale /= 1.1
z -= 1
else
scale *= 1.1
z += 1
end if
end while
reDraw()
end procedure
integer dx=0,dy=0 -- mouse down coords integer mx=0,my=0 -- mouse move/up coords
function mainHandler(integer id, integer msg, atom wParam, object lParam) integer x, y -- scratch vars atom scale10
if msg=WM_SIZE then -- (also activate/firsttime)
{{},{},x,y} = getClientRect(Main)
if firsttime or cWidth!=x or cHeight!=y then
scale *= cWidth/x
{cWidth, cHeight} = {x,y}
newDib()
firsttime = 0
end if
elsif msg=WM_PAINT then
copyDib(mainDC, 0, 0, mainDib)
if drawBox then
void = c_func(xRectangle, {mainDC, dx, dy, mx, my})
end if
if drawTime then
if drawTime=2 then
setText(Main,sprintf("Mandelbrot Set [generated in %gs]",time()-t0))
else
setText(Main,"Mandelbrot Set")
end if
drawTime -= 1
end if
elsif msg=WM_CHAR then
if wParam=VK_ESCAPE then
closeWindow(Main)
elsif wParam='+' then zoom(+1)
elsif wParam='-' then zoom(-1)
end if
elsif msg=WM_LBUTTONDOWN
or msg=WM_RBUTTONDOWN then
{dx,dy} = lParam
elsif msg=WM_MOUSEMOVE then
if and_bits(wParam,MK_LBUTTON) then
{mx,my} = lParam
-- minus dx,dy (see WM_LBUTTONUP)
-- DEV maybe a timer to redraw, but probably too slow...
-- (this is where we need a background worker thread,
-- ideally one we can direct to abandon what it is
-- currently doing and start work on new x,y instead)
elsif and_bits(wParam,MK_RBUTTON) then
{mx,my} = lParam
drawBox = 1
repaintWindow(Main,False)
end if
elsif msg=WM_MOUSEWHEEL then
wParam = floor(wParam/#10000)
if wParam>=#8000 then -- sign bit set
wParam-=#10000
end if
wParam = floor(wParam/120) -- (gives +/-1, usually)
zoom(wParam)
elsif msg=WM_LBUTTONUP then
{mx,my} = lParam
drawBox = 0
x0 += (dx-mx)*scale
y0 += (dy-my)*scale
reDraw()
elsif msg=WM_RBUTTONUP then
{mx,my} = lParam
drawBox = 0
if mx!=dx and my!=dy then
x0 += min(mx,dx)*scale
y0 += min(my,dy)*scale
scale *= (abs(mx-dx))/cHeight
reDraw()
end if
elsif msg=WM_KEYDOWN then
if wParam=VK_F1 then
Help()
elsif wParam=VK_F2 then
{x,y} = getWindowRect(Main)
void = c_func(xTrackPopupMenu, {iterHwnd,TPM_LEFTALIGN,x+20,y+40,0,mainHwnd,NULL})
elsif find(wParam,{VK_UP,VK_DOWN,VK_LEFT,VK_RIGHT}) then
drawBox = 0
scale10 = scale*10
if wParam=VK_UP then
y0 += scale10
elsif wParam=VK_DOWN then
y0 -= scale10
elsif wParam=VK_LEFT then
x0 += scale10
elsif wParam=VK_RIGHT then
x0 -= scale10
end if
reDraw()
end if
elsif msg=WM_COMMAND then
id = find(id,m50to1000)
if id!=0 then
iter = i50to1000[id]
reDraw()
end if
end if
return 0
end function setHandler({Main,mIter50,mIter100,mIter500,mIter1000}, routine_id("mainHandler"))
WinMain(Main,SW_NORMAL) void = deleteDib(0)</lang>
PHP
<lang PHP>$min_x=-2; $max_x=1; $min_y=-1; $max_y=1;
$dim_x=400; $dim_y=300;
$im = @imagecreate($dim_x, $dim_y)
or die("Cannot Initialize new GD image stream");
header("Content-Type: image/png"); $black_color = imagecolorallocate($im, 0, 0, 0); $white_color = imagecolorallocate($im, 255, 255, 255);
for($y=0;$y<=$dim_y;$y++) {
for($x=0;$x<=$dim_x;$x++) {
$c1=$min_x+($max_x-$min_x)/$dim_x*$x;
$c2=$min_y+($max_y-$min_y)/$dim_y*$y;
$z1=0; $z2=0;
for($i=0;$i<100;$i++) {
$new1=$z1*$z1-$z2*$z2+$c1;
$new2=2*$z1*$z2+$c2;
$z1=$new1;
$z2=$new2;
if($z1*$z1+$z2*$z2>=4) {
break;
}
}
if($i<100) {
imagesetpixel ($im, $x, $y, $white_color);
}
}
}
imagepng($im); imagedestroy($im); </lang>
Picat
Picat has not support for fancy graphics so it's plain 0/1 ASCII. Also, there's no built-in support for complex numbers. <lang Picat>go =>
Width = 90,
Height = 50,
Limit = 50,
foreach(Y in 1..Height)
P="",
foreach (X in 1..Width)
Z=complex(0,0),
C=complex(2.5*X/Width-2.0,2.0*Y/Height-1.0),
J = 0,
while (J < Limit, c_abs(Z)<2.0)
Z := c_add(c_mul(Z,Z),C),
J := J + 1
end,
if J == Limit then
P := P ++ "#"
else
P := P ++ "."
end
end,
printf("%s\n", P)
end,
nl.
% Operations on complex numbers complex(R,I) = [R,I]. c_add(X,Y) = complex(X[1]+Y[1],X[2]+Y[2]). c_mul(X,Y) = complex(X[1]*Y[1]-X[2]*Y[2],X[1]*Y[2]+X[2]*Y[1]). c_abs(X) = sqrt(X[1]*X[1]+X[2]*X[2]).</lang>
- Output:
.......................................................................................... .......................................................................................... ...................................................................##..................... ..................................................................##...................... ...............................................................########................... ..............................................................#########................... ...............................................................#######.................... ................................................................######.................... ..........................................................#.#.############................ ...................................................###..#####################............. ...................................................###########################..####...... .....................................................##############################....... ..................................................###############################......... ..............................................#.###################################....... ................................................####################################...... ................................................####################################.#.... ...............................................#######################################.... ...............................#...##.........#######################################..... .............................##########......#########################################.... ..............................############...#########################################.... ...........................################..#########################################.... ...........................#################.########################################..... ...........................#################.#######################################...... .......................############################################################....... #################################################################################......... .......................############################################################....... ...........................#################.#######################################...... ...........................#################.########################################..... ...........................################..#########################################.... ..............................############...#########################################.... .............................##########......#########################################.... ...............................#...##.........#######################################..... ...............................................#######################################.... ................................................####################################.#.... ................................................####################################...... ..............................................#.###################################....... ..................................................###############################......... .....................................................##############################....... ...................................................###########################..####...... ...................................................###..#####################............. ..........................................................#.#.############................ ................................................................######.................... ...............................................................#######.................... ..............................................................#########................... ...............................................................########................... ..................................................................##...................... ...................................................................##..................... .......................................................................................... .......................................................................................... .......................................................................#..................
PicoLisp
<lang PicoLisp>(scl 6)
(let Ppm (make (do 300 (link (need 400))))
(for (Y . Row) Ppm
(for (X . @) Row
(let (ZX 0 ZY 0 CX (*/ (- X 250) 1.0 150) CY (*/ (- Y 150) 1.0 150) C 570)
(while (and (> 4.0 (+ (*/ ZX ZX 1.0) (*/ ZY ZY 1.0))) (gt0 C))
(let Tmp (- (*/ ZX ZX 1.0) (*/ ZY ZY 1.0) (- CX))
(setq
ZY (+ (*/ 2 ZX ZY 1.0) CY)
ZX Tmp ) )
(dec 'C) )
(set (nth Ppm Y X) (list 0 C C)) ) ) )
(out "img.ppm"
(prinl "P6")
(prinl 400 " " 300)
(prinl 255)
(for Y Ppm (for X Y (apply wr X))) ) )</lang>
PostScript
<lang postscript>%!PS-Adobe-2.0 %%BoundingBox: 0 0 300 200 %%EndComments /origstate save def /ld {load def} bind def /m /moveto ld /g /setgray ld /dot { currentpoint 1 0 360 arc fill } bind def %%EndProlog % param /maxiter 200 def % complex manipulation /complex { 2 array astore } def /real { 0 get } def /imag { 1 get } def /cmul { /a exch def /b exch def
a real b real mul a imag b imag mul sub a real b imag mul a imag b real mul add 2 array astore
} def /cadd { aload pop 3 -1 roll aload pop
3 -1 roll add 3 1 roll add exch 2 array astore
} def /cconj { aload pop neg 2 array astore } def /cabs2 { dup cconj cmul 0 get} def % mandel 200 100 translate -200 1 100 { /x exch def
-100 1 100 { /y exch def
/z0 0.0 0.0 complex def
0 1 maxiter { /iter exch def
x 100 div y 100 div complex z0 z0 cmul cadd dup /z0 exch def cabs2 4 gt {exit} if
} for iter maxiter div g x y m dot } for
} for % showpage origstate restore %%EOF</lang>
PowerShell
<lang PowerShell> $x = $y = $i = $j = $r = -16 $colors = [Enum]::GetValues([System.ConsoleColor])
while(($y++) -lt 15) {
for($x=0; ($x++) -lt 84; Write-Host " " -BackgroundColor ($colors[$k -band 15]) -NoNewline)
{
$i = $k = $r = 0
do
{
$j = $r * $r - $i * $i -2 + $x / 25
$i = 2 * $r * $i + $y / 10
$r = $j
}
while (($j * $j + $i * $i) -lt 11 -band ($k++) -lt 111)
}
Write-Host
} </lang>
Processing
Click on an area to zoom in. Choose areas with multiple colors for interesting zooming. <lang java>double x, y, zr, zi, zr2, zi2, cr, ci, n; double zmx1, zmx2, zmy1, zmy2, f, di, dj; double fn1, fn2, fn3, re, gr, bl, xt, yt, i, j;
void setup() {
size(500, 500); di = 0; dj = 0; f = 10; fn1 = random(20); fn2 = random(20); fn3 = random(20); zmx1 = int(width / 4); zmx2 = 2; zmy1 = int(height / 4); zmy2 = 2;
}
void draw() {
if (i <= width) i++;
x = (i + di)/ zmx1 - zmx2;
for ( j = 0; j <= height; j++) {
y = zmy2 - (j + dj) / zmy1;
zr = 0;
zi = 0;
zr2 = 0;
zi2 = 0;
cr = x;
ci = y;
n = 1;
while (n < 200 && (zr2 + zi2) < 4) {
zi2 = zi * zi;
zr2 = zr * zr;
zi = 2 * zi * zr + ci;
zr = zr2 - zi2 + cr;
n++;
}
re = (n * fn1) % 255;
gr = (n * fn2) % 255;
bl = (n * fn3) % 255;
stroke((float)re, (float)gr, (float)bl);
point((float)i, (float)j);
}
}
void mousePressed() {
background(200); xt = mouseX; yt = mouseY; di = di + xt - float(width / 2); dj = dj + yt - float(height / 2); zmx1 = zmx1 * f; zmx2 = zmx2 * (1 / f); zmy1 = zmy1 * f; zmy2 = zmy2 * (1 / f); di = di * f; dj = dj * f; i = 0; j = 0;
}</lang>The sketch can be run online :
here.
Processing Python mode
Click on an area to zoom in. Choose areas with multiple colors for interesting zooming. <lang python>i = di = dj = 0 fn1, fn2, fn3 = random(20), random(20), random(20) f = 10
def setup():
global zmx1, zmx2, zmy1, zmy2 size(500, 500) zmx1 = int(width / 4) zmx2 = 2 zmy1 = int(height / 4) zmy2 = 2
def draw():
global i
if i <= width:
i += 1
x = float(i + di) / zmx1 - zmx2
for j in range(height + 1):
y = zmy2 - float(j + dj) / zmy1
zr = zi = zr2 = zi2 = 0
cr, ci = x, y
n = 1
while n < 200 and (zr2 + zi2) < 4:
zi2 = zi * zi
zr2 = zr * zr
zi = 2 * zi * zr + ci
zr = zr2 - zi2 + cr
n += 1
re = (n * fn1) % 255
gr = (n * fn2) % 255
bl = (n * fn3) % 255
stroke(re, gr, bl)
point(i, j)
def mousePressed():
global zmx1, zmx2, zmy1, zmy2, di, dj global i, j background(200) xt, yt = mouseX, mouseY di = di + xt - width / 2. dj = dj + yt - height / 2. zmx1 = zmx1 * f zmx2 = zmx2 * (1. / f) zmy1 = zmy1 * f zmy2 = zmy2 * (1. / f) di, dj = di * f, dj * f i = j = 0</lang>
Prolog
SWI-Prolog has a graphic interface XPCE : <lang Prolog>:- use_module(library(pce)).
mandelbrot :-
new(D, window('Mandelbrot Set')),
send(D, size, size(700, 650)),
new(Img, image(@nil, width := 700, height := 650, kind := pixmap)),
forall(between(0,699, I),
( forall(between(0,649, J),
( get_RGB(I, J, R, G, B),
R1 is (R * 256) mod 65536,
G1 is (G * 256) mod 65536,
B1 is (B * 256) mod 65536,
send(Img, pixel(I, J, colour(@default, R1, G1, B1))))))),
new(Bmp, bitmap(Img)),
send(D, display, Bmp, point(0,0)),
send(D, open).
get_RGB(X, Y, R, G, B) :-
CX is (X - 350) / 150, CY is (Y - 325) / 150, Iter = 570, compute_RGB(CX, CY, 0, 0, Iter, It), IterF is It \/ It << 15, R is IterF >> 16, Iter1 is IterF - R << 16, G is Iter1 >> 8, B is Iter1 - G << 8.
compute_RGB(CX, CY, ZX, ZY, Iter, IterF) :-
ZX * ZX + ZY * ZY < 4, Iter > 0, !, Tmp is ZX * ZX - ZY * ZY + CX, ZY1 is 2 * ZX * ZY + CY, Iter1 is Iter - 1, compute_RGB(CX, CY, Tmp, ZY1, Iter1, IterF).
compute_RGB(_CX, _CY, _ZX, _ZY, Iter, Iter).</lang>Example :
Python
Translation of the ruby solution <lang python># Python 3.0+ and 2.5+ try:
from functools import reduce
except:
pass
def mandelbrot(a):
return reduce(lambda z, _: z * z + a, range(50), 0)
def step(start, step, iterations):
return (start + (i * step) for i in range(iterations))
rows = (("*" if abs(mandelbrot(complex(x, y))) < 2 else " "
for x in step(-2.0, .0315, 80))
for y in step(1, -.05, 41))
print("\n".join("".join(row) for row in rows)) </lang>
A more "Pythonic" version of the code: <lang python> import math
def mandelbrot(z , c , n=40):
if abs(z) > 1000:
return float("nan")
elif n > 0:
return mandelbrot(z ** 2 + c, c, n - 1)
else:
return z ** 2 + c
print("\n".join(["".join(["#" if not math.isnan(mandelbrot(0, x + 1j * y).real) else " "
for x in [a * 0.02 for a in range(-80, 30)]])
for y in [a * 0.05 for a in range(-20, 20)]])
)
</lang>
Finally, we can also use Matplotlib to visualize the Mandelbrot set with Python:
<lang python>from pylab import * from numpy import NaN
def m(a): z = 0 for n in range(1, 100): z = z**2 + a if abs(z) > 2: return n return NaN
X = arange(-2, .5, .002) Y = arange(-1, 1, .002) Z = zeros((len(Y), len(X)))
for iy, y in enumerate(Y): print (iy, "of", len(Y)) for ix, x in enumerate(X): Z[iy,ix] = m(x + 1j * y)
imshow(Z, cmap = plt.cm.prism, interpolation = 'none', extent = (X.min(), X.max(), Y.min(), Y.max())) xlabel("Re(c)") ylabel("Im(c)") savefig("mandelbrot_python.svg") show()</lang>
Another Numpy version using masks to avoid (explicit) nested loops. Runs about 16x faster for the same resolution. <lang python>import matplotlib.pyplot as plt import numpy as np
npts = 300 max_iter = 100
X = np.linspace(-2, 1, 2 * npts) Y = np.linspace(-1, 1, npts)
- broadcast X to a square array
C = X[:, None] + 1J * Y
- initial value is always zero
Z = np.zeros_like(C)
exit_times = max_iter * np.ones(C.shape, np.int32) mask = exit_times > 0
for k in range(max_iter):
Z[mask] = Z[mask] * Z[mask] + C[mask] mask, old_mask = abs(Z) < 2, mask #use XOR to detect the area which has changed exit_times[mask ^ old_mask] = k
plt.imshow(exit_times.T,
cmap=plt.cm.prism,
extent=(X.min(), X.max(), Y.min(), Y.max()))
plt.show()</lang>
Actually the same, but without optimizations and therefore better suited for teaching. At first without NumPy, but already with complex numbers. <lang python>import matplotlib.pyplot as plt
def linspace(start, stop, num):
return [start + (stop - start) / (num - 1) * k for k in range(num)]
def zeros(rows, cols):
return [[0 for j in range(cols)] for i in range(rows)]
d, n = 200, 50 # pixel density & number of iterations
r = 2.5 # escape radius (must be greater than 2)
x = linspace(-2.5, 1.5, 4 * d + 1) y = linspace(-1.5, 1.5, 3 * d + 1)
T = zeros(len(y), len(x))
for i, b in enumerate(y):
for j, a in enumerate(x):
c, z = a + b * 1j, 0j
for k in range(n):
if abs(z) < r:
z = z ** 2 + c
T[i][j] = k + 1
plt.imshow(T, cmap=plt.cm.twilight_shifted) plt.savefig("Mandelbrot.png", dpi=250)</lang> At second with NumPy and complex matrices. <lang python>import numpy as np import matplotlib.pyplot as plt
d, n = 200, 50 # pixel density & number of iterations r = 2.5 # escape radius (must be greater than 2)
x = np.linspace(-2.5, 1.5, 4 * d + 1) y = np.linspace(-1.5, 1.5, 3 * d + 1)
A, B = np.meshgrid(x, y) C = A + B * 1j
Z = np.zeros_like(C) T = np.zeros(C.shape)
for k in range(n):
M = abs(Z) < r Z[M] = Z[M] ** 2 + C[M] T[M] = k + 1
plt.imshow(T, cmap=plt.cm.twilight_shifted) plt.savefig("Mandelbrot.png", dpi=250)</lang>
Normalized Iteration Count, Distance Estimation and Mercator Maps
The Mandelbrot set is printed with smooth colors. The e^(-|z|)-smoothing, normalized iteration count and exterior distance estimation algorithms are used with NumPy and complex matrices (see Javier Barrallo & Damien M. Jones: Coloring Algorithms for Dynamical Systems in the Complex Plane and Mikael Hvidtfeldt Christensen: Distance Estimated 3D Fractals (V): The Mandelbulb & Different DE Approximations). <lang python>import numpy as np import matplotlib.pyplot as plt
d, h = 800, 500 # pixel density (= image width) and image height n, r = 200, 500 # number of iterations and escape radius (r > 2)
x = np.linspace(0, 2, num=d+1) y = np.linspace(0, 2 * h / d, num=h+1)
A, B = np.meshgrid(x - 1, y - h / d) C = (2.0 - 1.0j) * (A + B * 1j) - 0.5
Z, dZ = np.zeros_like(C), np.zeros_like(C) S, T, D = np.zeros(C.shape), np.zeros(C.shape), np.zeros(C.shape)
for k in range(n):
M = abs(Z) < r S[M], T[M] = S[M] + np.exp(- abs(Z[M])), T[M] + 1 Z[M], dZ[M] = Z[M] ** 2 + C[M], 2 * Z[M] * dZ[M] + 1
plt.imshow(S ** 0.1, cmap=plt.cm.twilight_shifted) plt.savefig("Mandelbrot_set_1.png", dpi=200)
N = abs(Z) > r # normalized iteration count T[N] = T[N] - np.log2(np.log(abs(Z[N])) / np.log(r))
plt.imshow(T ** 0.1, cmap=plt.cm.twilight_shifted) plt.savefig("Mandelbrot_set_2.png", dpi=200)
N = abs(Z) > 2 # exterior distance estimation D[N] = 0.5 * np.log(abs(Z[N])) * abs(Z[N]) / abs(dZ[N])
plt.imshow(D ** 0.1, cmap=plt.cm.twilight_shifted) plt.savefig("Mandelbrot_set_3.png", dpi=200)</lang> A small change in the code above creates Mercator maps of the Mandelbrot set (see David Madore: Mandelbrot set images and videos and Anders Sandberg: Mercator Mandelbrot Maps). The maximum magnification is exp(2*pi*h/d) = exp(2*pi*5.5) = 535.5^5.5 = 10^15, which is also the maximum for 64-bit arithmetic. Note that Anders Sandberg uses a different scaling. He uses 10^(3*h/d) = 1000^(h/d) instead of exp(2*pi*h/d) = 535.5^(h/d), so his images appear somewhat compressed in comparison (but not much, because 1000^5 = 10^15 = 535.5^5.5). With the same pixel density and the same maximum magnification, the difference in height between the maps is only about 10 percent. By misusing a scatter plot, it is possible to create zoom images of the Mandelbrot set. To extend the zoom range beyond h > 5.5d, calculations must be performed using floating point numbers of arbitrary precision. This is currently not possible with NumPy, but can be implemented very easily in the Julia programming language (see the corresponding program example for the Julia language). <lang python>import numpy as np import matplotlib.pyplot as plt
d, h = 200, 1200 # pixel density (= image width) and image height n, r = 800, 1000 # number of iterations and escape radius (r > 2)
x = np.linspace(0, 2, num=d+1) y = np.linspace(0, 2 * h / d, num=h+1)
A, B = np.meshgrid(x * np.pi, y * np.pi) C = (- 8.0j) * np.exp((A + B * 1j) * 1j) - 0.7436636774 + 0.1318632144j
Z, dZ = np.zeros_like(C), np.zeros_like(C) D = np.zeros(C.shape)
for k in range(n):
M = abs(Z) < r Z[M], dZ[M] = Z[M] ** 2 + C[M], 2 * Z[M] * dZ[M] + 1
N = abs(Z) > 2 # exterior distance estimation D[N] = 0.5 * np.log(abs(Z[N])) * abs(Z[N]) / abs(dZ[N])
plt.imshow(D.T ** 0.1, cmap=plt.cm.nipy_spectral) plt.savefig("Mercator_Mandelbrot_map.png", dpi=200)
X, Y = C.real, C.imag # zoom images (adjust circle size 120 and zoom level 20 as needed) R, c, z = 120 * 2 / d * np.pi * np.exp(- B), min(d, h) + 1, max(0, h - d) // 20
fig, ax = plt.subplots(2, 2, figsize=(12, 12)) ax[0, 0].scatter(X[1*z:1*z+c], Y[1*z:1*z+c], s=R[0:c]**2, c=D[1*z:1*z+c]**0.5, cmap=plt.cm.nipy_spectral) ax[0, 1].scatter(X[2*z:2*z+c], Y[2*z:2*z+c], s=R[0:c]**2, c=D[2*z:2*z+c]**0.4, cmap=plt.cm.nipy_spectral) ax[1, 0].scatter(X[3*z:3*z+c], Y[3*z:3*z+c], s=R[0:c]**2, c=D[3*z:3*z+c]**0.3, cmap=plt.cm.nipy_spectral) ax[1, 1].scatter(X[4*z:4*z+c], Y[4*z:4*z+c], s=R[0:c]**2, c=D[4*z:4*z+c]**0.2, cmap=plt.cm.nipy_spectral) plt.savefig("Mercator_Mandelbrot_plot.png", dpi=100)</lang>
Python - "One liner"
<lang python># works on python 3.8 and above print(.join([.join([chr(round(36-2*(a:=0+0j,abs([a:=a*a+complex(x,y)if abs(a)<2 else(1.4+1.5j)for _ in range(100)][-1]))[1]))for x in map((lambda x:x/1000),range(-2000,500,20))])+'\n'for y in map((lambda x:x/1000),range(1000,-1000,-20))])) </lang>
A variant of the above reformatted for legibility,
and to conform to the useful convention of an 80 char maximum line width:
<lang python>Mandelbrot Set
print(.join(
[.join(
[chr(round(
36 - 2 * (
a := 0 + 0j,
abs([
a := a * a + complex(x, y)
if abs(a) < 2 else (
1.4 + 1.5j
) for _ in range(100)
][-1])
)[1]
)) for x in map(
lambda x:x / 1000,
range(-2000, 500, 20)
)]
) + '\n' for y in map(
lambda x:x / 1000,
range(1000, -1000, -20)
)]
))</lang>
R
<lang R>iterate.until.escape <- function(z, c, trans, cond, max=50, response=dwell) {
#we iterate all active points in the same array operation,
#and keeping track of which points are still iterating.
active <- seq_along(z)
dwell <- z
dwell[] <- 0
for (i in 1:max) {
z[active] <- trans(z[active], c[active]);
survived <- cond(z[active])
dwell[active[!survived]] <- i
active <- active[survived]
if (length(active) == 0) break
}
eval(substitute(response))
}
re = seq(-2, 1, len=500) im = seq(-1.5, 1.5, len=500) c <- outer(re, im, function(x,y) complex(real=x, imaginary=y)) x <- iterate.until.escape(array(0, dim(c)), c,
function(z,c)z^2+c, function(z)abs(z) <= 2,
max=100)
image(x)</lang>
Mandelbrot Set with R Animation
Modified Mandelbrot set animation by Jarek Tuszynski, PhD. (see: Wikipedia: R (programming_language) and R Tricks: Mandelbrot Set with R Animation) <lang R>#install.packages("caTools") # install external package (if missing) library(caTools) # external package providing write.gif function jet.colors <- colorRampPalette(c("red", "blue", "#007FFF", "cyan", "#7FFF7F",
"yellow", "#FF7F00", "red", "#7F0000"))
dx <- 800 # define width dy <- 600 # define height C <- complex(real = rep(seq(-2.5, 1.5, length.out = dx), each = dy),
imag = rep(seq(-1.5, 1.5, length.out = dy), dx))
C <- matrix(C, dy, dx) # reshape as square matrix of complex numbers Z <- 0 # initialize Z to zero X <- array(0, c(dy, dx, 20)) # initialize output 3D array for (k in 1:20) { # loop with 20 iterations
Z <- Z^2 + C # the central difference equation X[, , k] <- exp(-abs(Z)) # capture results
} write.gif(X, "Mandelbrot.gif", col = jet.colors, delay = 100)</lang>
Racket
<lang racket>
- lang racket
(require racket/draw)
(define (iterations a z i)
(define z′ (+ (* z z) a))
(if (or (= i 255) (> (magnitude z′) 2))
i
(iterations a z′ (add1 i))))
(define (iter->color i)
(if (= i 255)
(make-object color% "black")
(make-object color% (* 5 (modulo i 15)) (* 32 (modulo i 7)) (* 8 (modulo i 31)))))
(define (mandelbrot width height)
(define target (make-bitmap width height)) (define dc (new bitmap-dc% [bitmap target])) (for* ([x width] [y height]) (define real-x (- (* 3.0 (/ x width)) 2.25)) (define real-y (- (* 2.5 (/ y height)) 1.25)) (send dc set-pen (iter->color (iterations (make-rectangular real-x real-y) 0 0)) 1 'solid) (send dc draw-point x y)) (send target save-file "mandelbrot.png" 'png))
(mandelbrot 300 200) </lang>
Raku
(formerly Perl 6)
Variant of a Mandelbrot script from the Raku ecosystem. Produces a Portable Pixel Map to STDOUT. Redirect into a file to save it. Converted to a .png file for display here.
<lang perl6>constant @color_map = map ~*.comb(/../).map({:16($_)}), < 000000 0000fc 4000fc 7c00fc bc00fc fc00fc fc00bc fc007c fc0040 fc0000 fc4000 fc7c00 fcbc00 fcfc00 bcfc00 7cfc00 40fc00 00fc00 00fc40 00fc7c 00fcbc 00fcfc 00bcfc 007cfc 0040fc 7c7cfc 9c7cfc bc7cfc dc7cfc fc7cfc fc7cdc fc7cbc fc7c9c fc7c7c fc9c7c fcbc7c fcdc7c fcfc7c dcfc7c bcfc7c 9cfc7c 7cfc7c 7cfc9c 7cfcbc 7cfcdc 7cfcfc 7cdcfc 7cbcfc 7c9cfc b4b4fc c4b4fc d8b4fc e8b4fc fcb4fc fcb4e8 fcb4d8 fcb4c4 fcb4b4 fcc4b4 fcd8b4 fce8b4 fcfcb4 e8fcb4 d8fcb4 c4fcb4 b4fcb4 b4fcc4 b4fcd8 b4fce8 b4fcfc b4e8fc b4d8fc b4c4fc 000070 1c0070 380070 540070 700070 700054 700038 70001c 700000 701c00 703800 705400 707000 547000 387000 1c7000 007000 00701c 007038 007054 007070 005470 003870 001c70 383870 443870 543870 603870 703870 703860 703854 703844 703838 704438 705438 706038 707038 607038 547038 447038 387038 387044 387054 387060 387070 386070 385470 384470 505070 585070 605070 685070 705070 705068 705060 705058 705050 705850 706050 706850 707050 687050 607050 587050 507050 507058 507060 507068 507070 506870 506070 505870 000040 100040 200040 300040 400040 400030 400020 400010 400000 401000 402000 403000 404000 304000 204000 104000 004000 004010 004020 004030 004040 003040 002040 001040 202040 282040 302040 382040 402040 402038 402030 402028 402020 402820 403020 403820 404020 384020 304020 284020 204020 204028 204030 204038 204040 203840 203040 202840 2c2c40 302c40 342c40 3c2c40 402c40 402c3c 402c34 402c30 402c2c 40302c 40342c 403c2c 40402c 3c402c 34402c 30402c 2c402c 2c4030 2c4034 2c403c 2c4040 2c3c40 2c3440 2c3040 >;
constant MAX_ITERATIONS = 50; my $width = my $height = +(@*ARGS[0] // 31);
sub cut(Range $r, UInt $n where $n > 1) {
$r.min, * + ($r.max - $r.min) / ($n - 1) ... $r.max
}
my @re = cut(-2 .. 1/2, $height); my @im = cut( 0 .. 5/4, $width div 2 + 1) X* 1i;
sub mandelbrot(Complex $z is copy, Complex $c) {
for 1 .. MAX_ITERATIONS {
$z = $z*$z + $c; return $_ if $z.abs > 2;
} return 0;
}
say "P3"; say "$width $height"; say "255";
for @re -> $re {
put @color_map[|.reverse, |.[1..*]][^$width] given my @ = map &mandelbrot.assuming(0i, *), $re «+« @im;
}</lang>
Alternately, a more modern, faster version.
<lang perl6>use Image::PNG::Portable;
my ($w, $h) = 800, 800; my $out = Image::PNG::Portable.new: :width($w), :height($h);
my $maxIter = 150;
my @re = scale(-2.05 .. 1.05, $h); my @im = scale( -11/8 .. 11/8, $w) X* 1i;
for ^($w div 2) -> $x {
^$h .map: -> $y {
my $i = (mandelbrot( @re[$y] + @im[$x] ) / $maxIter) ** .25;
my @hsv = hsv2rgb($i, 1, ?$i).rotate;
$out.set: $x, $y, |@hsv;
$out.set: $w - 1 - $x, $y, |@hsv;
}
}
$out.write: 'Mandelbrot-set-perl6.png';
sub scale (Range $r,Int $n) { $r.min, * + ($r.max - $r.min) / ($n - 1) ... $r.max }
sub mandelbrot(Complex $c) {
my $z = $c;
for ^$maxIter {
$z = $z * $z + $c; .return if $z.abs > 2;
} 0
}
sub hsv2rgb ( $h, $s, $v ){
state %cache;
%cache{"$h|$s|$v"} //= do {
my $c = $v * $s;
my $x = $c * (1 - abs( (($h*6) % 2) - 1 ) );
my $m = $v - $c;
[(do given $h {
when 0..^1/6 { $c, $x, 0 }
when 1/6..^1/3 { $x, $c, 0 }
when 1/3..^1/2 { 0, $c, $x }
when 1/2..^2/3 { 0, $x, $c }
when 2/3..^5/6 { $x, 0, $c }
when 5/6..1 { $c, 0, $x }
} ).map: ((*+$m) * 255).Int]
}
}</lang> See Mandelbrot-set-perl6.png (offsite .png image)
REXX
version 1
This REXX version doesn't depend on the ASCII sequence of glyphs; an internal character string was used that mimics a part of the ASCII glyph sequence. <lang rexx>/*REXX program generates and displays a Mandelbrot set as an ASCII art character image.*/ @ = '>=<;:9876543210/.-,+*)(&%$#"!' /*the characters used in the display. */ Xsize = 59; minRE = -2; maxRE = +1; stepX = (maxRE-minRE) / Xsize Ysize = 21; minIM = -1; maxIM = +1; stepY = (maxIM-minIM) / Ysize
do y=0 for ysize; im=minIM + stepY*y
$=
do x=0 for Xsize; re=minRE + stepX*x; zr=re; zi=im
do n=0 for 30; a=zr**2; b=zi**2; if a+b>4 then leave
zi=zr*zi*2 + im; zr=a-b+re
end /*n*/
$=$ || substr(@, n+1, 1) /*append number (as a char) to $ string*/
end /*x*/
say $ /*display a line of character output.*/
end /*y*/ /*stick a fork in it, we're all done. */</lang>
output using the internal defaults:
>>>>>>=====<<<<<<<<<<<<<<<;;;;;;:::96032:;;;;<<<<========== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<======= >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= ><<<<;;;;;:::972456-567763 +9;;<<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >::988897735/ &89:;;;<<<<<< >;;;;;;::997564' ' 8:;;;<<<<<< ><;;;;;;::::9875& .3 *9;;;<<<<<< ><<<<;;;;;:::972456-567763 +9;;<<<<<<< >=<<<<<<<<;;;:599999999886 %78:;;<<<<<<= >>=<<<<<<<<<<<;;;::::::999752 *79:;<<<<<<== >>==<<<<<<<<<<<<<;;;;::::996. &2 45335:;<<<<<<=== >>>==<<<<<<<<<<<<<<<;;;;;;:98888764 5789999:;;<<<<<==== >>>>===<<<<<<<<<<<<<<<;;;;;;;::9974 (.9::::;;<<<<<====== >>>>>===<<<<<<<<<<<<<<<<;;;;;;;:::873*079::;;;;<<<<<=======
version 2
This REXX version uses glyphs that are "darker" (with a white background) around the output's peripheral. <lang rexx>/*REXX program generates and displays a Mandelbrot set as an ASCII art character image.*/ @ = '█▓▒░@9876543210=.-,+*)(·&%$#"!' /*the characters used in the display. */ Xsize = 59; minRE = -2; maxRE = +1; stepX = (maxRE-minRE) / Xsize Ysize = 21; minIM = -1; maxIM = +1; stepY = (maxIM-minIM) / Ysize
do y=0 for ysize; im=minIM + stepY*y
$=
do x=0 for Xsize; re=minRE + stepX*x; zr=re; zi=im
do n=0 for 30; a=zr**2; b=zi**2; if a+b>4 then leave
zi=zr*zi*2 + im; zr=a-b+re
end /*n*/
$=$ || substr(@, n+1, 1) /*append number (as a char) to $ string*/
end /*x*/
say $ /*display a line of character output.*/
end /*y*/ /*stick a fork in it, we're all done. */</lang>
output using the internal defaults:
██████▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░@@@96032@░░░░▒▒▒▒▓▓▓▓▓▓▓▓▓▓ █████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@@873*079@@░░░░▒▒▒▒▒▓▓▓▓▓▓▓ ████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@9974 (.9@@@@░░▒▒▒▒▒▓▓▓▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░@98888764 5789999@░░▒▒▒▒▒▓▓▓▓ ██▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░@@@@996. &2 45335@░▒▒▒▒▒▒▓▓▓ ██▓▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@999752 *79@░▒▒▒▒▒▒▓▓ █▓▒▒▒▒▒▒▒▒░░░@599999999886 %78@░░▒▒▒▒▒▒▓ █▒▒▒▒░░░░░@@@972456-567763 +9░░▒▒▒▒▒▒▒ █▒░░░░░░@@@@9875& .3 *9░░░▒▒▒▒▒▒ █░░░░░░@@997564· · 8@░░░▒▒▒▒▒▒ █@@988897735= &89@░░░▒▒▒▒▒▒ █@@988897735= &89@░░░▒▒▒▒▒▒ █░░░░░░@@997564· · 8@░░░▒▒▒▒▒▒ █▒░░░░░░@@@@9875& .3 *9░░░▒▒▒▒▒▒ █▒▒▒▒░░░░░@@@972456-567763 +9░░▒▒▒▒▒▒▒ █▓▒▒▒▒▒▒▒▒░░░@599999999886 %78@░░▒▒▒▒▒▒▓ ██▓▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@999752 *79@░▒▒▒▒▒▒▓▓ ██▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░@@@@996. &2 45335@░▒▒▒▒▒▒▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░@98888764 5789999@░░▒▒▒▒▒▓▓▓▓ ████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@9974 (.9@@@@░░▒▒▒▒▒▓▓▓▓▓▓ █████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@@873*079@@░░░░▒▒▒▒▒▓▓▓▓▓▓▓
version 3
This REXX version produces a larger output (it uses the full width of the terminal screen (less one), and the height is one-half of the width. <lang rexx>/*REXX program generates and displays a Mandelbrot set as an ASCII art character image.*/ @ = '█▓▒░@9876543210=.-,+*)(·&%$#"!' /*the characters used in the display. */ parse arg Xsize Ysize . /*obtain optional arguments from the CL*/ if Xsize== then Xsize=linesize() - 1 /*X: the (usable) linesize (minus 1).*/ if Ysize== then Ysize=Xsize%2 + (Xsize//2==1) /*Y: half the linesize (make it even).*/ minRE = -2; maxRE = +1; stepX = (maxRE-minRE) / Xsize minIM = -1; maxIM = +1; stepY = (maxIM-minIM) / Ysize
do y=0 for ysize; im=minIM + stepY*y
$=
do x=0 for Xsize; re=minRE + stepX*x; zr=re; zi=im
do n=0 for 30; a=zr**2; b=zi**2; if a+b>4 then leave
zi=zr*zi*2 + im; zr=a-b+re
end /*n*/
$=$ || substr(@, n+1, 1) /*append number (as a char) to $ string*/
end /*x*/
say $ /*display a line of character output.*/
end /*y*/ /*stick a fork in it, we're all done. */</lang>
output using the internal defaults:
████████▓▓▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░@@@@985164(9@░░░░░░▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓▓▓▓▓ ███████▓▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@@@98763=5799@░░░░░░▒▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓▓▓ ███████▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@@98763.-2789@@@░░░░░░▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓▓ ██████▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@@985 2. 1448@@@@░░░░░░▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓ █████▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@999874 *=79@@@@@░░░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓▓▓ █████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@999998873 17899@@@@@░░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓▓ ████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@98888888764 #4678899999@@░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓ ████▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@@@9343,665 322= =215357888709@░░▒▒▒▒▒▒▒▒▓▓▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░@@@@@9986 + 32 ,56554)79@░▒▒▒▒▒▒▒▒▒▓▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░@@@@@@@999863 + 2· ",59@░░▒▒▒▒▒▒▒▒▓▓▓▓ ███▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@@@@@9998763 $ 379@@░▒▒▒▒▒▒▒▒▒▓▓▓ ██▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@@@@@999986.2 $ +689@@░░▒▒▒▒▒▒▒▒▒▓▓ ██▓▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@689999@@@99999887 05789@░░▒▒▒▒▒▒▒▒▒▓▓ ██▒▒▒▒▒▒▒▒▒▒░░░░░@@9717888877888888763. 2558@░░░▒▒▒▒▒▒▒▒▒▓ █▓▒▒▒▒▒▒░░░░░░░@@@996)566761467777762 4@░░░▒▒▒▒▒▒▒▒▒▓ █▓▒▒▒▒░░░░░░░░@@@@99763 & 42&..366651· .68@░░░░▒▒▒▒▒▒▒▒▓ █▒▒▒░░░░░░░░@@@@@@98864* ) $ 343 259@░░░░▒▒▒▒▒▒▒▒▒ █▒▒░░░░░░░░@@@@@@988753. 11 #9@░░░░▒▒▒▒▒▒▒▒▒ █▒░░░░░░░░@@@@@9877650 - 289@░░░░▒▒▒▒▒▒▒▒▒ █░░░░░░░░@9999887%413+ % &69@@░░░░▒▒▒▒▒▒▒▒▒ █░@@@@@89999888763 % ( 389@@░░░░▒▒▒▒▒▒▒▒▒ █@@99872676676422 5789@@░░░░▒▒▒▒▒▒▒▒▒ █@@99872676676422 5789@@░░░░▒▒▒▒▒▒▒▒▒ █░@@@@@89999888763 % ( 389@@░░░░▒▒▒▒▒▒▒▒▒ █░░░░░░░░@9999887%413+ % &69@@░░░░▒▒▒▒▒▒▒▒▒ █▒░░░░░░░░@@@@@9877650 - 289@░░░░▒▒▒▒▒▒▒▒▒ █▒▒░░░░░░░░@@@@@@988753. 11 #9@░░░░▒▒▒▒▒▒▒▒▒ █▒▒▒░░░░░░░░@@@@@@98864* ) $ 343 259@░░░░▒▒▒▒▒▒▒▒▒ █▓▒▒▒▒░░░░░░░░@@@@99763 & 42&..366651· .68@░░░░▒▒▒▒▒▒▒▒▓ █▓▒▒▒▒▒▒░░░░░░░@@@996)566761467777762 4@░░░▒▒▒▒▒▒▒▒▒▓ ██▒▒▒▒▒▒▒▒▒▒░░░░░@@9717888877888888763. 2558@░░░▒▒▒▒▒▒▒▒▒▓ ██▓▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@689999@@@99999887 05789@░░▒▒▒▒▒▒▒▒▒▓▓ ██▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@@@@@999986.2 $ +689@@░░▒▒▒▒▒▒▒▒▒▓▓ ███▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░@@@@@@@@@@9998763 $ 379@@░▒▒▒▒▒▒▒▒▒▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░@@@@@@@999863 + 2· ",59@░░▒▒▒▒▒▒▒▒▓▓▓▓ ███▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░@@@@@9986 + 32 ,56554)79@░▒▒▒▒▒▒▒▒▒▓▓▓▓ ████▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░@@@@9343,665 322= =215357888709@░░▒▒▒▒▒▒▒▒▓▓▓▓▓ ████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@98888888764 #4678899999@@░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓ █████▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@999998873 17899@@@@@░░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓▓ █████▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@999874 *=79@@@@@░░░░▒▒▒▒▒▒▒▒▓▓▓▓▓▓▓▓ ██████▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@@985 2. 1448@@@@░░░░░░▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓ ███████▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░░@@@@98763.-2789@@@░░░░░░▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓▓ ███████▓▓▓▓▓▓▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒░░░░░░░░░@@@@98763=5799@░░░░░░▒▒▒▒▒▒▒▓▓▓▓▓▓▓▓▓▓▓▓
This REXX program makes use of linesize REXX program (or BIF) which is used to determine the screen width (or linesize) of the terminal (console).
The LINESIZE.REX REXX program is included here ──► LINESIZE.REX.
Ring
<lang ring> load "guilib.ring"
new qapp
{
win1 = new qwidget() {
setwindowtitle("Mandelbrot set")
setgeometry(100,100,500,500)
label1 = new qlabel(win1) {
setgeometry(10,10,400,400)
settext("")
}
new qpushbutton(win1) {
setgeometry(200,400,100,30)
settext("draw")
setclickevent("draw()")
}
show()
}
exec()
}
func draw
p1 = new qpicture()
color = new qcolor() {
setrgb(0,0,255,255)
}
pen = new qpen() {
setcolor(color)
setwidth(1)
}
new qpainter() {
begin(p1)
setpen(pen)
x1=300 y1=250
i1=-1 i2=1 r1=-2 r2=1
s1=(r2-r1)/x1 s2=(i2-i1)/y1
for y=0 to y1
i3=i1+s2*y
for x=0 to x1
r3=r1+s1*x z1=r3 z2=i3
for n=0 to 30
a=z1*z1 b=z2*z2
if a+b>4 exit ok
z2=2*z1*z2+i3 z1=a-b+r3
next
if n != 31 drawpoint(x,y) ok
next
next
endpaint()
}
label1 { setpicture(p1) show() }
</lang>
Output:
Ruby
Text only, prints an 80-char by 41-line depiction. Found here. <lang ruby>require 'complex'
def mandelbrot(a)
Array.new(50).inject(0) { |z,c| z*z + a }
end
(1.0).step(-1,-0.05) do |y|
(-2.0).step(0.5,0.0315) do |x| print mandelbrot(Complex(x,y)).abs < 2 ? '*' : ' ' end puts
end</lang>
Uses Raster graphics operations/Ruby <lang ruby># frozen_string_literal: true
require_relative 'raster_graphics'
class RGBColour
def self.mandel_colour(i) self.new( 16*(i % 15), 32*(i % 7), 8*(i % 31) ) end
end
class Pixmap
def self.mandelbrot(width, height)
mandel = Pixmap.new(width,height)
pb = ProgressBar.new(width) if $DEBUG
width.times do |x|
height.times do |y|
x_ish = Float(x - width*11/15) / (width/3)
y_ish = Float(y - height/2) / (height*3/10)
mandel[x,y] = RGBColour.mandel_colour(mandel_iters(x_ish, y_ish))
end
pb.update(x) if $DEBUG
end
pb.close if $DEBUG
mandel
end
def self.mandel_iters(cx,cy)
x = y = 0.0
count = 0
while Math.hypot(x,y) < 2 and count < 255
x, y = (x**2 - y**2 + cx), (2*x*y + cy)
count += 1
end
count
end
end
Pixmap.mandelbrot(300,300).save('mandel.ppm')</lang>
JRubyArt is a port of processing to ruby <lang ruby>
- frozen_string_literal: true
def setup
sketch_title 'Mandelbrot' load_pixels no_loop
end
def draw
grid(900, 600) do |x, y|
const = Complex(
map1d(x, (0...900), (-3..1.5)), map1d(y, (0...600), (-1.5..1.5))
)
pixels[x + y * 900] = color(
constrained_map(mandel(const, 20), (5..20), (255..0))
)
end
update_pixels
end
def mandel(z, max)
score = 0 const = z while score < max # z = z^2 + c z *= z z += const break if z.abs > 2
score += 1 end score
end
def settings
size(900, 600)
end </lang>
Rust
Dependencies: image, num-complex <lang rust>extern crate image; extern crate num_complex;
use std::fs::File; use num_complex::Complex;
fn main() {
let max_iterations = 256u16; let img_side = 800u32; let cxmin = -2f32; let cxmax = 1f32; let cymin = -1.5f32; let cymax = 1.5f32; let scalex = (cxmax - cxmin) / img_side as f32; let scaley = (cymax - cymin) / img_side as f32;
// Create a new ImgBuf let mut imgbuf = image::ImageBuffer::new(img_side, img_side);
// Calculate for each pixel
for (x, y, pixel) in imgbuf.enumerate_pixels_mut() {
let cx = cxmin + x as f32 * scalex;
let cy = cymin + y as f32 * scaley;
let c = Complex::new(cx, cy);
let mut z = Complex::new(0f32, 0f32);
let mut i = 0;
for t in 0..max_iterations {
if z.norm() > 2.0 {
break;
}
z = z * z + c;
i = t;
}
*pixel = image::Luma([i as u8]); }
// Save image
imgbuf.save("fractal.png").unwrap();
}</lang>
Sass/SCSS
<lang coffeescript> $canvasWidth: 200; $canvasHeight: 200; $iterations: 20; $xCorner: -2; $yCorner: -1.5; $zoom: 3; $data: ()!global; @mixin plot ($x,$y,$count){
$index: ($y * $canvasWidth + $x) * 4; $r: $count * -12 + 255; $g: $count * -12 + 255; $b: $count * -12 + 255; $data: append($data, $x + px $y + px 0 rgb($r,$g,$b), comma)!global;
}
@for $x from 1 to $canvasWidth {
@for $y from 1 to $canvasHeight {
$count: 0;
$size: 0;
$cx: $xCorner + (($x * $zoom) / $canvasWidth);
$cy: $yCorner + (($y * $zoom) / $canvasHeight);
$zx: 0;
$zy: 0;
@while $count < $iterations and $size <= 4 {
$count: $count + 1;
$temp: ($zx * $zx) - ($zy * $zy);
$zy: (2 * $zx * $zy) + $cy;
$zx: $temp + $cx;
$size: ($zx * $zx) + ($zy * $zy);
}
@include plot($x, $y, $count); }
} .set {
height: 1px; width: 1px; position: absolute; top: 50%; left: 50%; transform: translate($canvasWidth*0.5px, $canvasWidth*0.5px); box-shadow: $data;
}
</lang>
Scala
Uses RgbBitmap from Basic Bitmap Storage task and Complex number class from this programming task. <lang scala>import org.rosettacode.ArithmeticComplex._ import java.awt.Color
object Mandelbrot {
def generate(width:Int =600, height:Int =400)={
val bm=new RgbBitmap(width, height)
val maxIter=1000
val xMin = -2.0
val xMax = 1.0
val yMin = -1.0
val yMax = 1.0
val cx=(xMax-xMin)/width
val cy=(yMax-yMin)/height
for(y <- 0 until bm.height; x <- 0 until bm.width){
val c=Complex(xMin+x*cx, yMin+y*cy)
val iter=itMandel(c, maxIter, 4)
bm.setPixel(x, y, getColor(iter, maxIter))
}
bm
}
def itMandel(c:Complex, imax:Int, bailout:Int):Int={
var z=Complex()
for(i <- 0 until imax){
z=z*z+c;
if(z.abs > bailout) return i
}
imax;
}
def getColor(iter:Int, max:Int):Color={
if (iter==max) return Color.BLACK
var c=3*math.log(iter)/math.log(max-1.0)
if(c<1) new Color((255*c).toInt, 0, 0)
else if(c<2) new Color(255, (255*(c-1)).toInt, 0)
else new Color(255, 255, (255*(c-2)).toInt)
}
}</lang> Read–eval–print loop <lang scala>import scala.swing._ import javax.swing.ImageIcon val imgMandel=Mandelbrot.generate() val mainframe=new MainFrame(){title="Test"; visible=true
contents=new Label(){icon=new ImageIcon(imgMandel.image)}
}</lang>
Scheme
This implementation writes an image of the Mandelbrot set to a plain pgm file. The set itself is drawn in white, while the exterior is drawn in black. <lang scheme>(define x-centre -0.5) (define y-centre 0.0) (define width 4.0) (define i-max 800) (define j-max 600) (define n 100) (define r-max 2.0) (define file "out.pgm") (define colour-max 255) (define pixel-size (/ width i-max)) (define x-offset (- x-centre (* 0.5 pixel-size (+ i-max 1)))) (define y-offset (+ y-centre (* 0.5 pixel-size (+ j-max 1))))
(define (inside? z)
(define (*inside? z-0 z n)
(and (< (magnitude z) r-max)
(or (= n 0)
(*inside? z-0 (+ (* z z) z-0) (- n 1)))))
(*inside? z 0 n))
(define (boolean->integer b)
(if b colour-max 0))
(define (pixel i j)
(boolean->integer
(inside?
(make-rectangular (+ x-offset (* pixel-size i))
(- y-offset (* pixel-size j))))))
(define (plot)
(with-output-to-file file
(lambda ()
(begin (display "P2") (newline)
(display i-max) (newline)
(display j-max) (newline)
(display colour-max) (newline)
(do ((j 1 (+ j 1))) ((> j j-max))
(do ((i 1 (+ i 1))) ((> i i-max))
(begin (display (pixel i j)) (newline))))))))
(plot)</lang>
Scratch
Seed7
$ include "seed7_05.s7i";
include "float.s7i";
include "complex.s7i";
include "draw.s7i";
include "keybd.s7i";
# Display the Mandelbrot set, that are points z[0] in the complex plane
# for which the sequence z[n+1] := z[n] ** 2 + z[0] (n >= 0) is bounded.
# Since this program is computing intensive it should be compiled with
# hi comp -O2 mandelbr
const integer: pix is 200;
const integer: max_iter is 256;
var array color: colorTable is max_iter times black;
const func integer: iterate (in complex: z0) is func
result
var integer: iter is 1;
local
var complex: z is complex.value;
begin
z := z0;
while sqrAbs(z) < 4.0 and # not diverged
iter < max_iter do # not converged
z *:= z;
z +:= z0;
incr(iter);
end while;
end func;
const proc: displayMandelbrotSet (in complex: center, in float: zoom) is func
local
var integer: x is 0;
var integer: y is 0;
var complex: z0 is complex.value;
begin
for x range -pix to pix do
for y range -pix to pix do
z0 := center + complex(flt(x) * zoom, flt(y) * zoom);
point(x + pix, y + pix, colorTable[iterate(z0)]);
end for;
end for;
end func;
const proc: main is func
local
const integer: num_pix is 2 * pix + 1;
var integer: col is 0;
begin
screen(num_pix, num_pix);
clear(curr_win, black);
KEYBOARD := GRAPH_KEYBOARD;
for col range 1 to pred(max_iter) do
colorTable[col] := color(65535 - (col * 5003) mod 65535,
(col * 257) mod 65535,
(col * 2609) mod 65535);
end for;
displayMandelbrotSet(complex(-0.75, 0.0), 1.3 / flt(pix));
DRAW_FLUSH;
readln(KEYBOARD);
end func;
Original source: [2]
SenseTalk
<lang sensetalk>put 0 into oReal # Real origin put 0 into oImag # Imaginary origin put 0.5 into mag # Magnification
put oReal - .8 / mag into leftReal put oImag + .5 / mag into topImag put 1 / 200 / mag into inc
put [ (0,255,255), # aqua (0,0,255), # blue (255,0,255), # fuchsia (128,128,128), # gray (0,128,0), # green (0,255,0), # lime (128,0,0), # maroon (0,0,128), # navy (128,128,0), # olive (128,0,128), # purple (255,0,0), # red (192,192,192), # silver (0,128,128), # teal (255,255,255), # white (255,255,0) # yellow ] into colors
put "mandelbrot.ppm" into myFile
open file myFile for writing write "P3" & return to file myFile # PPM file magic number write "320 200" & return to file myFile # Width and height write "255" & return to file myFile # Max value in color channels
put topImag into cImag repeat with each item in 1 .. 200 put leftReal into cReal repeat with each item in 1 .. 320 put 0 into zReal put 0 into zImag put 0 into count put 0 into size repeat at least once until size > 2 or count = 100 put zReal squared + zImag squared * -1 into newZreal put zReal * zImag + zReal * zImag into newZimag put newZreal + cReal into zReal put newZimag + cImag into zImag put sqrt(zReal squared + zImag squared) into size add 1 to count end repeat if size > 2 then # Outside the set - colorize put item count mod 15 + 1 of colors into color write color joined by " " to file myFile write return to file myFile else # Inside the set - black write "0 0 0" & return to file myFile end if add inc to cReal end repeat subtract inc from cImag end repeat
close file myFile </lang>
SequenceL
SequenceL Code for Computing and Coloring:
<lang sequencel>import <Utilities/Complex.sl>;
import <Utilities/Sequence.sl>;
import <Utilities/Math.sl>;
COLOR_STRUCT ::= (R: int(0), G: int(0), B: int(0)); rgb(r(0), g(0), b(0)) := (R: r, G: g, B: b);
RESULT_STRUCT ::= (FinalValue: Complex(0), Iterations: int(0)); makeResult(val(0), iters(0)) := (FinalValue: val, Iterations: iters);
zSquaredOperation(startingNum(0), currentNum(0)) :=
complexAdd(startingNum, complexMultiply(currentNum, currentNum));
zSquared(minX(0), maxX(0), resolutionX(0), minY(0), maxY(0), resolutionY(0), maxMagnitude(0), maxIters(0))[Y,X] :=
let
stepX := (maxX - minX) / resolutionX;
stepY := (maxY - minY) / resolutionY;
currentX := X * stepX + minX;
currentY := Y * stepY + minY;
in
operateUntil(zSquaredOperation, makeComplex(currentX, currentY), makeComplex(currentX, currentY), maxMagnitude, 0, maxIters)
foreach Y within 0 ... (resolutionY - 1),
X within 0 ... (resolutionX - 1);
operateUntil(operation(0), startingNum(0), currentNum(0), maxMagnitude(0), currentIters(0), maxIters(0)) :=
let
operated := operation(startingNum, currentNum);
in
makeResult(currentNum, maxIters) when currentIters >= maxIters
else
makeResult(currentNum, currentIters) when complexMagnitude(currentNum) >= maxMagnitude
else
operateUntil(operation, startingNum, operated, maxMagnitude, currentIters + 1, maxIters);
//region Smooth Coloring
COLOR_COUNT := size(colorSelections);
colorRange := range(0, 255, 1);
colors :=
let
first[i] := rgb(0, 0, i) foreach i within colorRange;
second[i] := rgb(i, i, 255) foreach i within colorRange;
third[i] := rgb(255, 255, i) foreach i within reverse(colorRange);
fourth[i] := rgb(255, i, 0) foreach i within reverse(colorRange);
fifth[i] := rgb(i, 0, 0) foreach i within reverse(colorRange);
red[i] := rgb(i, 0, 0) foreach i within colorRange;
redR[i] := rgb(i, 0, 0) foreach i within reverse(colorRange);
green[i] := rgb(0, i, 0) foreach i within colorRange;
greenR[i] :=rgb(0, i, 0) foreach i within reverse(colorRange);
blue[i] := rgb(0, 0, i) foreach i within colorRange;
blueR[i] := rgb(0, 0, i) foreach i within reverse(colorRange);
in
//red ++ redR ++ green ++ greenR ++ blue ++ blueR;
first ++ second ++ third ++ fourth ++ fifth;
//first ++ fourth;
colorSelections := range(1, size(colors), 30);
getSmoothColorings(zSquaredResult(2), maxIters(0))[Y,X] :=
let
current := zSquaredResult[Y,X];
zn := complexMagnitude(current.FinalValue);
nu := ln(ln(zn) / ln(2)) / ln(2);
result := abs(current.Iterations + 1 - nu);
index := floor(result);
rem := result - index;
color1 := colorSelections[(index mod COLOR_COUNT) + 1];
color2 := colorSelections[((index + 1) mod COLOR_COUNT) + 1];
in
rgb(0, 0, 0) when current.Iterations = maxIters
else
colors[color1] when color2 < color1
else
colors[floor(linearInterpolate(color1, color2, rem))];
linearInterpolate(v0(0), v1(0), t(0)) := (1 - t) * v0 + t * v1;
//endregion</lang>
C++ Driver Code:
<lang c>#include "SL_Generated.h"
- include "../../../ThirdParty/CImg/CImg.h"
using namespace std; using namespace cimg_library;
int main(int argc, char ** argv) {
int cores = 0;
Sequence<Sequence<_sl_RESULT_STRUCT> > computeResult; Sequence<Sequence<_sl_COLOR_STRUCT> > colorResult;
sl_init(cores);
int maxIters = 1000; int imageWidth = 1920; int imageHeight = 1200; double maxMag = 256;
double xmin = -2.5; double xmax = 1.0; double ymin = -1.0; double ymax = 1.0;
CImg<unsigned char> visu(imageWidth, imageHeight, 1, 3); CImgDisplay draw_disp(visu, "Mandelbrot Fractal in SequenceL");
bool redraw = true;
SLTimer t;
double computeTime;
double colorTime;
double renderTime;
while(!draw_disp.is_closed())
{
if(redraw)
{
redraw = false;
t.start();
sl_zSquared(xmin, xmax, imageWidth, ymin, ymax, imageHeight, maxMag, maxIters, cores, computeResult);
t.stop();
computeTime = t.getTime();
t.start();
sl_getSmoothColorings(computeResult, maxIters, cores, colorResult);
t.stop();
colorTime = t.getTime();
t.start();
visu.fill(0);
for(int i = 1; i <= colorResult.size(); i++)
{
for(int j = 1; j <= colorResult[i].size(); j++)
{
visu(j-1,i-1,0,0) = colorResult[i][j].R;
visu(j-1,i-1,0,1) = colorResult[i][j].G;
visu(j-1,i-1,0,2) = colorResult[i][j].B;
}
}
visu.display(draw_disp);
t.stop();
renderTime = t.getTime();
draw_disp.set_title("X:[%f, %f] Y:[%f, %f] | Mandelbrot Fractal in SequenceL | Compute Time: %f | Color Time: %f | Render Time: %f | Total FPS: %f", xmin, xmax, ymin, ymax, cores, computeTime, colorTime, renderTime, 1 / (computeTime + colorTime + renderTime));
}
draw_disp.wait();
double xdiff = (xmax - xmin);
double ydiff = (ymax - ymin);
double xcenter = ((1.0 * draw_disp.mouse_x()) / imageWidth) * xdiff + xmin;
double ycenter = ((1.0 * draw_disp.mouse_y()) / imageHeight) * ydiff + ymin;
if(draw_disp.button()&1)
{
redraw = true;
xmin = xcenter - (xdiff / 4);
xmax = xcenter + (xdiff / 4);
ymin = ycenter - (ydiff / 4);
ymax = ycenter + (ydiff / 4);
}
else if(draw_disp.button()&2)
{
redraw = true;
xmin = xcenter - xdiff;
xmax = xcenter + xdiff;
ymin = ycenter - ydiff;
ymax = ycenter + ydiff;
}
}
sl_done();
return 0;
}</lang>
- Output:
Sidef
<lang ruby>func mandelbrot(z) {
var c = z
{ z = (z*z + c)
z.abs > 2 && return true
} * 20
return false
} for y range(1, -1, -0.05) {
for x in range(-2, 0.5, 0.0315) {
print(mandelbrot(x + y.i) ? ' ' : '#')
}
print "\n"
}</lang>
Simula
<lang simula>BEGIN
REAL XCENTRE, YCENTRE, WIDTH, RMAX, XOFFSET, YOFFSET, PIXELSIZE; INTEGER N, IMAX, JMAX, COLOURMAX; TEXT FILENAME;
CLASS COMPLEX(RE,IM); REAL RE,IM;;
REF(COMPLEX) PROCEDURE ADD(A,B); REF(COMPLEX) A,B;
ADD :- NEW COMPLEX(A.RE + B.RE, A.IM + B.IM);
REF(COMPLEX) PROCEDURE SUB(A,B); REF(COMPLEX) A,B;
SUB :- NEW COMPLEX(A.RE - B.RE, A.IM - B.IM);
REF(COMPLEX) PROCEDURE MUL(A,B); REF(COMPLEX) A,B;
MUL :- NEW COMPLEX(A.RE * B.RE - A.IM * B.IM,
A.RE * B.IM + A.IM * B.RE);
REF(COMPLEX) PROCEDURE DIV(A,B); REF(COMPLEX) A,B;
BEGIN
REAL TMP;
TMP := B.RE * B.RE + B.IM * B.IM;
DIV :- NEW COMPLEX((A.RE * B.RE + A.IM * B.IM) / TMP,
(A.IM * B.RE - A.RE * B.IM) / TMP);
END DIV;
REF(COMPLEX) PROCEDURE RECTANGULAR(RE,IM); REAL RE,IM;
RECTANGULAR :- NEW COMPLEX(RE,IM);
REAL PROCEDURE MAGNITUDE(CX); REF(COMPLEX) CX;
MAGNITUDE := SQRT(CX.RE**2 + CX.IM**2);
BOOLEAN PROCEDURE INSIDEP(Z); REF(COMPLEX) Z;
BEGIN
BOOLEAN PROCEDURE INSIDE(Z0,Z,N); REAL N; REF(COMPLEX) Z,Z0;
INSIDE := MAGNITUDE(Z) < RMAX
AND THEN N = 0 OR ELSE INSIDE(Z0, ADD(Z0,MUL(Z,Z)), N-1);
INSIDEP := INSIDE(Z, NEW COMPLEX(0,0), N);
END INSIDEP;
INTEGER PROCEDURE BOOL2INT(B); BOOLEAN B;
BOOL2INT := IF B THEN COLOURMAX ELSE 0;
INTEGER PROCEDURE PIXEL(I,J); INTEGER I,J;
PIXEL := BOOL2INT(INSIDEP(RECTANGULAR(XOFFSET + PIXELSIZE * I,
YOFFSET - PIXELSIZE * J)));
PROCEDURE PLOT;
BEGIN
REF (OUTFILE) OUTF;
INTEGER J,I;
OUTF :- NEW OUTFILE(FILENAME);
OUTF.OPEN(BLANKS(132));
OUTF.OUTTEXT("P2"); OUTF.OUTIMAGE;
OUTF.OUTINT(IMAX,0); OUTF.OUTIMAGE;
OUTF.OUTINT(JMAX,0); OUTF.OUTIMAGE;
OUTF.OUTINT(COLOURMAX,0); OUTF.OUTIMAGE;
FOR J := 1 STEP 1 UNTIL JMAX DO
BEGIN
FOR I := 1 STEP 1 UNTIL IMAX DO
BEGIN
OUTF.OUTINT(PIXEL(I,J),0);
OUTF.OUTIMAGE;
END;
END;
OUTF.CLOSE;
END PLOT;
XCENTRE := -0.5;
YCENTRE := 0.0;
WIDTH := 4.0;
IMAX := 800;
JMAX := 600;
N := 100;
RMAX := 2.0;
FILENAME :- "out.pgm";
COLOURMAX := 255;
PIXELSIZE := WIDTH / IMAX;
XOFFSET := XCENTRE - (0.5 * PIXELSIZE * (IMAX + 1));
YOFFSET := YCENTRE + (0.5 * PIXELSIZE * (JMAX + 1));
OUTTEXT("OUTPUT WILL BE WRITTEN TO ");
OUTTEXT(FILENAME);
OUTIMAGE;
PLOT;
END;</lang>
Spin
<lang spin>con
_clkmode = xtal1+pll16x _clkfreq = 80_000_000
xmin=-8601 ' int(-2.1*4096) xmax=2867 ' int( 0.7*4096)
ymin=-4915 ' int(-1.2*4096) ymax=4915 ' int( 1.2*4096)
maxiter=25
obj
ser : "FullDuplexSerial"
pub main | c,cx,cy,dx,dy,x,y,x2,y2,iter
ser.start(31, 30, 0, 115200)
dx:=(xmax-xmin)/79 dy:=(ymax-ymin)/24
cy:=ymin
repeat while cy=<ymax
cx:=xmin
repeat while cx=<xmax
x:=0
y:=0
x2:=0
y2:=0
iter:=0
repeat while iter=<maxiter and x2+y2=<16384
y:=((x*y)~>11)+cy
x:=x2-y2+cx
iter+=1
x2:=(x*x)~>12
y2:=(y*y)~>12
cx+=dx
ser.tx(iter+32)
cy+=dy
ser.str(string(13,10))
waitcnt(_clkfreq+cnt) ser.stop</lang>
- Output:
!!!!!!!!!!!!!!!"""""""""""""####################################""""""""""""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$$%'+)%%%$$$$$#####"""""""""""
!!!!!!!!!!!"""""""#######################$$$$$$$$%%%&&(+,)++&%$$$$$$######""""""
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*5:/+('&%%$$$$$$#######"""
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''),:::::::,'&%%%%%$$$$########
!!!!!!!"""####################$$$$$$$$%%%&'())((())*,::::::/+))('&&&&)'%$$######
!!!!!!""###################$$$$$%%%%%%&&&'+.:::/::::::::::::::::/++:..93%%$#####
!!!!!"################$$$%%%%%%%%%%&&&&'),+2:::::::::::::::::::::::::1(&&%$$####
!!!!"##########$$$$$%%&(-(''''''''''''(*,5::::::::::::::::::::::::::::+)-&%$$###
!!!!####$$$$$$$$%%%%%&'(*-:1.+.:-4+))**:::::::::::::::::::::::::::::::4-(&%$$$##
!!!!#$$$$$$$$$%%%%%%'''++.6:::::::::8/0::::::::::::::::::::::::::::::::3(%%$$$$#
!!!#$$$$$$$%&&&&''()/-5.5::::::::::::::::::::::::::::::::::::::::::::::'&%%$$$$#
!!!(**+/+:523/80/46::::::::::::::::::::::::::::::::::::::::::::::::4+)'&&%%$$$$#
!!!#$$$$$$$%&&&&''().-2.:::::::::::::::::::::::::::::::::::::::::::::::'&%%$$$$#
!!!!#$$$$$$$$$%%%%%&'''/,.7::::::::::/0::::::::::::::::::::::::::::::::0'%%$$$$#
!!!!####$$$$$$$$%%%%%&'(*-:2.,/:-5+))**:::::::::::::::::::::::::::::::4+(&%$$$##
!!!!"##########$$$$$%%&(-(''''(''''''((*,4:::::::::::::::::::::::::::4+).&%$$###
!!!!!"################$$$%%%%%%%%%%&&&&'):,4:::::::::::::::::::::::::/('&%%$####
!!!!!!""##################$$$$$$%%%%%%&&&'*.:::0::::::::::::::::1,,://9)%%$#####
!!!!!!!"""####################$$$$$$$$%%%&(())((()**-::::::/+)))'&&&')'%$$######
!!!!!!!!""""#####################$$$$$$$$$$%%%&&&''(,:::::::+'&&%%%%%$$$########
!!!!!!!!!"""""#######################$$$$$$$$$$%%%%&')*7:0+('&%%%$$$$$#######"""
!!!!!!!!!!!"""""""######################$$$$$$$$$%%%&&(+-).*&%$$$$$$######""""""
!!!!!!!!!!!!!"""""""""#######################$$$$$$%%'3(%%%$$$$$######""""""""""
!!!!!!!!!!!!!!!""""""""""""#####################################""""""""""""""""
SPL
<lang spl>w,h = #.scrsize() sfx = -2.5; sfy = -2*h/w; fs = 4/w
- .aaoff()
> y, 1...h
> x, 1...w fx = sfx + x*fs; fy = sfy + y*fs #.drawpoint(x,y,color(fx,fy):3) <
< color(x,y)=
zr = x; zi = y; n = 0; maxn = 150 > zr*zr+zi*zi<4 & n<maxn zrn = zr*zr-zi*zi+x; zin = 2*zr*zi+y zr = zrn; zi = zin; n += 1 < ? n=maxn, <= 0,0,0 <= #.hsv2rgb(n/maxn*360,1,1):3
.</lang>
Swift
Using the Swift Numerics package, as well as the C library Quick 'N Dirty BMP imported in Swift.
<lang swift>import Foundation import Numerics import QDBMP
public typealias Color = (red: UInt8, green: UInt8, blue: UInt8)
public class BitmapDrawer {
public let imageHeight: Int public let imageWidth: Int
var grid: Color?
private let bmp: OpaquePointer
public init(height: Int, width: Int) {
self.imageHeight = height
self.imageWidth = width
self.grid = Color?(repeating: [Color?](repeating: nil, count: height), count: width)
self.bmp = BMP_Create(UInt(width), UInt(height), 24)
checkError() }
deinit {
BMP_Free(bmp)
}
private func checkError() {
let err = BMP_GetError()
guard err == BMP_STATUS(0) else {
fatalError("\(err)")
}
}
public func save(to path: String = "~/Desktop/out.bmp") {
for x in 0..<imageWidth {
for y in 0..<imageHeight {
guard let color = grid[x][y] else { continue }
BMP_SetPixelRGB(bmp, UInt(x), UInt(y), color.red, color.green, color.blue)
checkError()
}
}
(path as NSString).expandingTildeInPath.withCString {s in
BMP_WriteFile(bmp, s)
}
}
public func setPixel(x: Int, y: Int, to color: Color?) {
grid[x][y] = color
}
}
let imageSize = 10_000 let canvas = BitmapDrawer(height: imageSize, width: imageSize) let maxIterations = 256 let cxMin = -2.0 let cxMax = 1.0 let cyMin = -1.5 let cyMax = 1.5 let scaleX = (cxMax - cxMin) / Double(imageSize) let scaleY = (cyMax - cyMin) / Double(imageSize)
for x in 0..<imageSize {
for y in 0..<imageSize {
let cx = cxMin + Double(x) * scaleX
let cy = cyMin + Double(y) * scaleY
let c = Complex(cx, cy) var z = Complex(0.0, 0.0) var i = 0
for t in 0..<maxIterations {
if z.magnitude > 2 {
break
}
z = z * z + c
i = t
}
canvas.setPixel(x: x, y: y, to: Color(red: UInt8(i), green: UInt8(i), blue: UInt8(i))) }
}
canvas.save()</lang>
Tcl
This code makes extensive use of Tk's built-in photo image system, which provides a 32-bit RGBA plotting surface that can be then quickly drawn in any number of places in the application. It uses a computational color scheme that was easy to code... <lang tcl>package require Tk
proc mandelIters {cx cy} {
set x [set y 0.0]
for {set count 0} {hypot($x,$y) < 2 && $count < 255} {incr count} {
set x1 [expr {$x*$x - $y*$y + $cx}]
set y1 [expr {2*$x*$y + $cy}]
set x $x1; set y $y1
}
return $count
} proc mandelColor {iter} {
set r [expr {16*($iter % 15)}]
set g [expr {32*($iter % 7)}]
set b [expr {8*($iter % 31)}]
format "#%02x%02x%02x" $r $g $b
} image create photo mandel -width 300 -height 300
- Build picture in strips, updating as we go so we have "progress" monitoring
- Also set the cursor to tell the user to wait while we work.
pack [label .mandel -image mandel -cursor watch] update for {set x 0} {$x < 300} {incr x} {
for {set y 0} {$y < 300} {incr y} {
set i [mandelIters [expr {($x-220)/100.}] [expr {($y-150)/90.}]]
mandel put [mandelColor $i] -to $x $y
}
update
} .mandel configure -cursor {}</lang>
Plain TeX
The pst-fractal package includes a Mandelbrot set drawn by emitting PostScript code (using PSTricks), so the actual work done in the printer or PostScript interpreter.
<lang TeX>\input pst-fractal \psfractal[type=Mandel,xWidth=14cm,yWidth=12cm,maxIter=30,dIter=20] (-2.5,-1.5)(1,1.5) \end</lang>
The coordinates are a rectangle in the complex plane to draw, scaled up to xWidth,yWidth.
More iterations with maxIter is higher resolution but slower.
dIter is a scale factor for the colours.
LaTeX
The pstricks-examples package which is samples from the PSTricks book includes similar for LaTeX (25-02-6.ltx and 33-02-6.ltx).
The PGF shadings library includes a Mandelbrot set.
In PGF 3.0 the calculations are done in PostScript code emitted, so the output size is small but it only does 10 iterations so is very low resolution.
<lang LaTeX>\documentclass{minimal} \usepackage{tikz} \usetikzlibrary{shadings} \begin{document} \begin{tikzpicture}
\shade[shading=Mandelbrot set] (0,0) rectangle (4,4);
\end{tikzpicture} \end{document}</lang>
LuaTeX
LuaLaTeX plus pgfplots code can be found at http://texwelt.de/wissen/fragen/3960/fraktale-mit-pgfplots.
The calculations are done by inline Lua code and the resulting bitmap shown with a PGF plot.
TI-83 BASIC
Based on the BASIC Version. Due to the TI-83's lack of power, it takes around 2 hours to complete at 16 iterations. <lang ti83b>PROGRAM:MANDELBR
- Input "ITER. ",D
- For(A,Xmin,Xmax,ΔX)
- For(B,Ymin,Ymax,ΔY)
- 0→X
- 0→Y
- 0→I
- D→M
- While X^2+Y^2≤4 and I<M
- X^2-Y^2+A→R
- 2XY+B→Y
- R→X
- I+1→I
- End
- If I≠M
- Then
- I→C
- Else
- 0→C
- End
- If C<1
- Pt-On(A,B)
- End
- End
- End
</lang>
TXR
Creates same mandelbrot.pgm file.
<lang txrlisp>(defvar x-centre -0.5) (defvar y-centre 0.0) (defvar width 4.0) (defvar i-max 800) (defvar j-max 600) (defvar n 100) (defvar r-max 2.0) (defvar file "mandelbrot.pgm") (defvar colour-max 255) (defvar pixel-size (/ width i-max)) (defvar x-offset (- x-centre (* 0.5 pixel-size (+ i-max 1)))) (defvar y-offset (+ y-centre (* 0.5 pixel-size (+ j-max 1))))
- with-output-to-file macro
(defmacro with-output-to-file (name . body)
^(let ((*stdout* (open-file ,name "w")))
(unwind-protect (progn ,*body) (close-stream *stdout*))))
- complex number library
(defmacro cplx (x y) ^(cons ,x ,y)) (defmacro re (c) ^(car ,c)) (defmacro im (c) ^(cdr ,c))
(defsymacro c0 '(0 . 0))
(macro-time
(defun with-cplx-expand (specs body)
(tree-case specs
(((re im expr) . rest)
^(tree-bind (,re . ,im) ,expr ,(with-cplx-expand rest body)))
(() (tree-case body
((a b . rest) ^(progn ,a ,b ,*rest))
((a) a)
(x (error "with-cplx: invalid body ~s" body))))
(x (error "with-cplx: bad args ~s" x)))))
(defmacro with-cplx (specs . body)
(with-cplx-expand specs body))
(defun c+ (x y)
(with-cplx ((a b x) (c d y)) (cplx (+ a c) (+ b d))))
(defun c* (x y)
(with-cplx ((a b x) (c d y)) (cplx (- (* a c) (* b d)) (+ (* b c) (* a d)))))
(defun modulus (z)
(with-cplx ((a b z)) (sqrt (+ (* a a) (* b b)))))
- Mandelbrot routines
(defun inside-p (z0 : (z c0) (n n))
(and (< (modulus z) r-max)
(or (zerop n)
(inside-p z0 (c+ (c* z z) z0) (- n 1)))))
(defmacro int-bool (b)
^(if ,b colour-max 0))
(defun pixel (i j)
(int-bool
(inside-p
(cplx (+ x-offset (* pixel-size i))
(- y-offset (* pixel-size j))))))
- Mandelbrot loop and output
(defun plot ()
(with-output-to-file file
(format t "P2\n~s\n~s\n~s\n" i-max j-max colour-max)
(each ((j (range 1 j-max)))
(each ((i (range 1 i-max)))
(format *stdout* "~s " (pixel i j)))
(put-line "" *stdout*))))
(plot)</lang>
uBasic/4tH
uBasic does not support floating point calculations, so fixed point arithmetic is used, with Value 10000 representing 1.0. The Mandelbrot image is drawn using ASCII characters 1-9 to show number of iterations. Iteration count 10 or more is represented with '@'. To compensate the aspect ratio of the font, step sizes in x and y directions are different. <lang>A =-21000 ' Left Edge = -2.1 B = 15000 ' Right Edge = 1.5 C = 15000 ' Top Edge = 1.5 D =-15000 ' Bottom Edge = -1.5 E = 200 ' Max Iteration Depth F = 350 ' X Step Size G = 750 ' Y Step Size
For L = C To D Step -G ' Y0
For K = A To B-1 Step F ' X0
V = 0 ' Y
U = 0 ' X
I = 32 ' Char To Be Displayed
For O = 0 To E-1 ' Iteration
X = (U/10 * U) / 1000 ' X*X
Y = (V/10 * V) / 1000 ' Y*Y
If (X + Y > 40000)
I = 48 + O ' Print Digit 0...9
If (O > 9) ' If Iteration Count > 9,
I = 64 ' Print '@'
Endif
Break
Endif
Z = X - Y + K ' Temp = X*X - Y*Y + X0
V = (U/10 * V) / 500 + L ' Y = 2*X*Y + Y0
U = Z ' X = Temp
Next
Gosub I ' Ins_char(I)
Next
Print
Next
End
' Translate number to ASCII
32 Print " "; : Return 48 Print "0"; : Return 49 Print "1"; : Return 50 Print "2"; : Return 51 Print "3"; : Return 52 Print "4"; : Return 53 Print "5"; : Return 54 Print "6"; : Return 55 Print "7"; : Return 56 Print "8"; : Return 57 Print "9"; : Return 64 Print "@"; : Return</lang> Output:
1111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333322222222222222222222222222222222222222 1111111111111222222222233333333333333333333333344444457655544443333332222222222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 11111111111222222333333333333333333333334444444445555678@@@@7654444443333332222222222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98655544444433333332222222222222222222222222 1111111122222333333333333333333333344444444445556668@@@ @@@@76555544444333333322222222222222222222222 1111111122233333333333333333333344444444445566667778@@ @987666555544433333333222222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@87777@95443333333322222222222222222222 1111112233333333333333333334444455555556678@@ @@ @@@@@@@8544333333333222222222222222222 1111122333333333333333334445555555555666789@@@ @86554433333333322222222222222222 1111123333333333333444466666555556666778@@@@ @@87655443333333332222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65443333333332222222222222222 1111333344444444455556679@@@@@@@@@@@99@@@ @@765444333333333222222222222222 1111334444444455555567789@@ @@@@ @855444333333333222222222222222 11114444444455555668@99@@@ @ @@655444433333333322222222222222 11134555556666677789@@@ @ @86655444433333333322222222222222 111 @@876555444433333333322222222222222 11134555556666677789@@@ @ @86655444433333333322222222222222 11114444444455555668@99@@@ @ @@655444433333333322222222222222 1111334444444455555567789@@ @@@@ @855444333333333222222222222222 1111333344444444455556679@@@@@@@@@@@99@@@ @@765444333333333222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65443333333332222222222222222 1111123333333333333444466666555556666778@@@@ @@87655443333333332222222222222222 1111122333333333333333334445555555555666789@@@ @86554433333333322222222222222222 1111112233333333333333333334444455555556678@@ @@ @@@@@@@8544333333333222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@87777@95443333333322222222222222222222 1111111122233333333333333333333344444444445566667778@@ @987666555544433333333222222222222222222222 1111111122222333333333333333333333344444444445556668@@@ @@@@76555544444333333322222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98655544444433333332222222222222222222222222 11111111111222222333333333333333333333334444444445555678@@@@7654444443333332222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 1111111111111222222222233333333333333333333333344444457655544443333332222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333322222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211 1111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111 0 OK, 0:1726
UNIX Shell
<lang bash>((xmin=-8601)) # int(-2.1*4096) ((xmax=2867)) # int( 0.7*4096)
((ymin=-4915)) # int(-1.2*4096) ((ymax=4915)) # int( 1.2*4096)
((maxiter=30))
((dx=(xmax-xmin)/72)) ((dy=(ymax-ymin)/24))
C='0123456789' ((lC=${#C}))
for((cy=ymax;cy>=ymin;cy-=dy)) ; do for((cx=xmin;cx<=xmax;cx+=dx)) ; do ((x=0,y=0,x2=0,y2=0)) for((iter=0;iter<maxiter && x2+y2<=16384;iter++)) ; do ((y=((x*y)>>11)+cy,x=x2-y2+cx,x2=(x*x)>>12,y2=(y*y)>>12)) done ((c=iter%lC)) echo -n ${C:$c:1} done echo done</lang>
- Output:
1111111111111222222222222333333333333333333333333333333333222222222222222 1111111111112222222233333333333333333333344444456015554444333332222222222 1111111111222222333333333333333333333444444445556704912544444433333222222 1111111112222333333333333333333333444444444555678970508655544444333333222 1111111222233333333333333333333444444444556667807000002076555544443333333 1111112223333333333333333333444444455577898889016000003099766662644333333 1111122333333333333333334444455555566793000800000000000000931045875443333 1111123333333333333344455555555566668014000000000000000000000009865544333 1111233333333344445568277777777777880600000000000000000000000009099544433 1111333344444445555678041513450199023000000000000000000000000000807544433 1112344444445555556771179000000000410000000000000000000000000000036544443 1114444444566667782404400000000000000000000000000000000000000000775544443 1119912160975272040000000000000000000000000000000000000000000219765544443 1114444444566667792405800000000000000000000000000000000000000000075544443 1113344444445555556773270000000000500000000000000000000000000000676544443 1111333344444445555678045623255199020000000000000000000000000000707544433 1111233333333444445568177777877777881500000000000000000000000009190544433 1111123333333333333344455555555566668126000000000000000000000009865544333 1111122333333333333333334444455555566793000100000000000000941355975443333 1111112223333333333333333334444444455588908889016000003099876670654433333 1111111222233333333333333333334444444445556667800000002976555554443333333 1111111112222333333333333333333333444444444555679060608655544444333333222 1111111111222222333333333333333333333444444445556702049544444433333222222 1111111111112222222233333333333333333333344444456205554444333333222222222 1111111111111222222222222333333333333333333333333333333333222222222222222
Vedit macro language
Vedit macro language does not support floating point calculations, so fixed point arithmetic is used, with Value 10000 representing 1.0. The Mandelbrot image is drawn using ASCII characters 1-9 to show number of iterations. Iteration count 10 or more is represented with '@'. To compensate the aspect ratio of the font, step sizes in x and y directions are different. <lang vedit>#1 =-21000 // left edge = -2.1
- 2 = 15000 // right edge = 1.5
- 3 = 15000 // top edge = 1.5
- 4 =-15000 // bottom edge = -1.5
- 5 = 200 // max iteration depth
- 6 = 350 // x step size
- 7 = 750 // y step size
Buf_Switch(Buf_Free) for (#12 = #3; #12 > #4; #12 -= #7) { // y0
for (#11 = #1; #11 < #2; #11 += #6) { // x0
#22 = 0 // y
#21 = 0 // x
#9 = ' ' // char to be displayed
for (#15 = 0; #15 < #5; #15++) { // iteration
#31 = (#21/10 * #21) / 1000 // x*x
#32 = (#22/10 * #22) / 1000 // y*y
if (#31 + #32 > 40000) {
#9 = '0' + #15 // print digit 0...9
if (#15 > 9) { // if iteration count > 9,
#9 = '@' // print '@'
}
break
}
#33 = #31 - #32 + #11 // temp = x*x - y*y + x0
#22 = (#21/10 * #22) / 500 + #12 // y = 2*x*y + y0
#21 = #33 // x = temp
}
Ins_Char(#9)
}
Ins_Newline
} BOF </lang>
- Output:
1111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333322222222222222222222222222222222222222 1111111111111222222222233333333333333333333333344444457655544443333332222222222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 11111111111222222333333333333333333333334444444445555678@@@@7654444443333332222222222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98655544444433333332222222222222222222222222 1111111122222333333333333333333333344444444445556668@@@ @@@@76555544444333333322222222222222222222222 1111111122233333333333333333333344444444445566667778@@ @987666555544433333333222222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@87777@95443333333322222222222222222222 1111112233333333333333333334444455555556678@@ @@ @@@@@@@8544333333333222222222222222222 1111122333333333333333334445555555555666789@@@ @86554433333333322222222222222222 1111123333333333333444466666555556666778@@@@ @@87655443333333332222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65443333333332222222222222222 1111333344444444455556679@@@@@@@@@@@99@@@ @@765444333333333222222222222222 1111334444444455555567789@@ @@@@ @855444333333333222222222222222 11114444444455555668@99@@@ @ @@655444433333333322222222222222 11134555556666677789@@@ @ @86655444433333333322222222222222 111 @@876555444433333333322222222222222 11134555556666677789@@@ @ @86655444433333333322222222222222 11114444444455555668@99@@@ @ @@655444433333333322222222222222 1111334444444455555567789@@ @@@@ @855444333333333222222222222222 1111333344444444455556679@@@@@@@@@@@99@@@ @@765444333333333222222222222222 111123333333344444455568@887789@8777788@@@ @@@@65443333333332222222222222222 1111123333333333333444466666555556666778@@@@ @@87655443333333332222222222222222 1111122333333333333333334445555555555666789@@@ @86554433333333322222222222222222 1111112233333333333333333334444455555556678@@ @@ @@@@@@@8544333333333222222222222222222 111111122333333333333333333333444444455556@@@@@99@@@@@@ @@@@@@87777@95443333333322222222222222222222 1111111122233333333333333333333344444444445566667778@@ @987666555544433333333222222222222222222222 1111111122222333333333333333333333344444444445556668@@@ @@@@76555544444333333322222222222222222222222 1111111112222223333333333333333333333444444444455556789@@@@98655544444433333332222222222222222222222222 11111111111222222333333333333333333333334444444445555678@@@@7654444443333332222222222222222222222222222 1111111111112222222233333333333333333333333444444445567@@6665444444333333222222222222222222222222222222 1111111111111222222222233333333333333333333333344444457655544443333332222222222222222222222222222222222 1111111111111112222222222222333333333333333333333333333333333333322222222222222222222222222222222222222 1111111111111111222222222222222222233333333333333333333333222222222222222222222222222222222222222222222 1111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222 1111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211
Wren
<lang ecmascript>import "graphics" for Canvas, Color import "dome" for Window
var MaxIters = 570 var Zoom = 150
class MandelbrotSet {
construct new(width, height) {
Window.title = "Mandelbrot Set"
Window.resize(width, height)
Canvas.resize(width, height)
_w = width
_h = height
}
init() {
createMandelbrot()
}
createMandelbrot() {
for (x in 0..._w) {
for (y in 0..._h) {
var zx = 0
var zy = 0
var cX = (x - 400) / Zoom
var cY = (y - 300) / Zoom
var i = MaxIters
while (zx * zx + zy * zy < 4 && i > 0) {
var tmp = zx * zx - zy * zy + cX
zy = 2 * zx * zy + cY
zx = tmp
i = i - 1
}
var r = i * 255 / MaxIters
Canvas.pset(x, y, Color.rgb(r, r, r))
}
}
}
update() {}
draw(alpha) {}
}
var Game = MandelbrotSet.new(800, 600)</lang>
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations int X, Y, \screen coordinates of current point
Cnt; \iteration counter
real Cx, Cy, \coordinates scaled to +/-2 range
Zx, Zy, \complex accumulator
Temp; \temporary scratch
[SetVid($112); \set 640x480x24 graphics mode for Y:= 0 to 480-1 do \for all points on the screen...
for X:= 0 to 640-1 do
[Cx:= (float(X)/640.0 - 0.5) * 4.0; \range: -2.0 to +2.0
Cy:= (float(Y-240)/240.0) * 1.5; \range: -1.5 to +1.5
Cnt:= 0; Zx:= 0.0; Zy:= 0.0; \initialize
loop [if Zx*Zx + Zy*Zy > 2.0 then \Z heads toward infinity
[Point(X, Y, Cnt<<21+Cnt<<10+Cnt<<3); \set color of pixel to
quit; \ rate it approached infinity
]; \move on to next point
Temp:= Zx*Zy;
Zx:= Zx*Zx - Zy*Zy + Cx; \calculate next iteration of Z
Zy:= 2.0*Temp + Cy;
Cnt:= Cnt+1; \count number of iterations
if Cnt >= 1000 then quit; \assume point is in Mandelbrot
]; \ set and leave it colored black
];
X:= ChIn(1); \wait for keystroke SetVid($03); \restore normal text display ]</lang>
- Output:
XSLT
The fact that you can create an image of the Mandelbrot Set with XSLT is sometimes under-appreciated. However, it has been discussed extensively on the internet so is best reproduced here, and the code can be executed directly in your browser at that site.
<lang xml> <?xml version="1.0" encoding="UTF-8"?> <xsl:stylesheet version="1.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
<xsl:output method="html" indent="no"
doctype-public="-//W3C//DTD HTML 4.01//EN" doctype-system="http://www.w3.org/TR/REC-html40/strict.dtd" />
<xsl:template match="/fractal">
<html> <head> <title>XSLT fractal</title> <style type="text/css">
body { color:#55F; background:#000 } pre { font-family:monospace; font-size:7px } pre span { background:<xsl:value-of select="background" /> }
</style> </head> <body>
Copyright © 1992,2007 Joel Yliluoma (<a href="http://iki.fi/bisqwit/">http://iki.fi/bisqwit/</a>)
XSLT fractal
<xsl:call-template name="bisqwit-mandelbrot" />
</body> </html>
</xsl:template>
<xsl:template name="bisqwit-mandelbrot"
><xsl:call-template name="bisqwit-mandelbrot-line"> <xsl:with-param name="y" select="y/min"/> </xsl:call-template
></xsl:template>
<xsl:template name="bisqwit-mandelbrot-line"
><xsl:param name="y" /><xsl:call-template name="bisqwit-mandelbrot-column"> <xsl:with-param name="x" select="x/min"/> <xsl:with-param name="y" select="$y"/> </xsl:call-template ><xsl:if test="$y < y/max" >
<xsl:call-template name="bisqwit-mandelbrot-line"> <xsl:with-param name="y" select="$y + y/step"/> </xsl:call-template ></xsl:if
></xsl:template>
<xsl:template name="bisqwit-mandelbrot-column"
><xsl:param name="x" /><xsl:param name="y" /><xsl:call-template name="bisqwit-mandelbrot-slot"> <xsl:with-param name="x" select="$x" /> <xsl:with-param name="y" select="$y" /> <xsl:with-param name="zr" select="$x" /> <xsl:with-param name="zi" select="$y" /> </xsl:call-template ><xsl:if test="$x < x/max" ><xsl:call-template name="bisqwit-mandelbrot-column"> <xsl:with-param name="x" select="$x + x/step"/> <xsl:with-param name="y" select="$y" /> </xsl:call-template ></xsl:if
></xsl:template>
<xsl:template name="bisqwit-mandelbrot-slot" ><xsl:param name="x"
/><xsl:param name="y"
/><xsl:param name="zr"
/><xsl:param name="zi"
/><xsl:param name="iter" select="0"
/><xsl:variable name="zrsqr" select="($zr * $zr)"
/><xsl:variable name="zisqr" select="($zi * $zi)"
/><xsl:choose>
<xsl:when test="(4*scale*scale >= $zrsqr + $zisqr) and (maxiter > $iter+1)"
><xsl:call-template name="bisqwit-mandelbrot-slot">
<xsl:with-param name="x" select="$x" />
<xsl:with-param name="y" select="$y" />
<xsl:with-param name="zi" select="(2 * $zr * $zi) div scale + $y" />
<xsl:with-param name="zr" select="($zrsqr - $zisqr) div scale + $x" />
<xsl:with-param name="iter" select="$iter + 1" />
</xsl:call-template
></xsl:when>
<xsl:otherwise
><xsl:variable name="magnitude" select="magnitude[@value=$iter]"
/><xsl:value-of select="$magnitude/symbol"
/></xsl:otherwise>
</xsl:choose
></xsl:template>
</xsl:stylesheet> </lang>
Z80 Assembly
<lang Z80>
- Compute a Mandelbrot set on a simple Z80 computer.
- Porting this program to another Z80 platform should be easy and straight-
- forward
- The only dependencies on my homebrew machine are the system-calls
- used to print strings and characters. These calls are performed by loading
- IX with the number of the system-call and performing an RST 08. To port this
- program to another operating system just replace these system-calls with
- the appropriate versions. Only three system-calls are used in the following
- _crlf
- Prints a CR/LF, _puts: Prints a 0-terminated string (the adress of
- which is expected in HL), and _putc
- Print a single character which is
- expected in A. RST 0 give control back to the monitor.
- include "mondef.asm"
org ram_start
scale equ 256 ; Do NOT change this - the
; arithmetic routines rely on
; this scaling factor! :-)
divergent equ scale * 4
ld hl, welcome ; Print a welcome message
ld ix, _puts
rst 08
- for (y = <initial_value> ; y <= y_end; y += y_step)
- {
outer_loop ld hl, (y_end) ; Is y <= y_end?
ld de, (y)
and a ; Clear carry
sbc hl, de ; Perform the comparison
jp m, mandel_end ; End of outer loop reached
- for (x = x_start; x <= x_end; x += x_step)
- {
ld hl, (x_start) ; x = x_start
ld (x), hl
inner_loop ld hl, (x_end) ; Is x <= x_end?
ld de, (x)
and a
sbc hl, de
jp m, inner_loop_end ; End of inner loop reached
- z_0 = z_1 = 0;
ld hl, 0
ld (z_0), hl
ld (z_1), hl
- for (iteration = iteration_max; iteration; iteration--)
- {
ld a, (iteration_max)
ld b, a
iteration_loop push bc ; iteration -> stack
- z2 = (z_0 * z_0 - z_1 * z_1) / SCALE;
ld de, (z_1) ; Compute DE HL = z_1 * z_1
ld bc, de
call mul_16
ld (z_0_square_low), hl ; z_0 ** 2 is needed later again
ld (z_0_square_high), de
ld de, (z_0) ; Compute DE HL = z_0 * z_0
ld bc, de
call mul_16
ld (z_1_square_low), hl ; z_1 ** 2 will be also needed
ld (z_1_square_high), de
and a ; Compute subtraction
ld bc, (z_0_square_low)
sbc hl, bc
ld (scratch_0), hl ; Save lower 16 bit of result
ld hl, de
ld bc, (z_0_square_high)
sbc hl, bc
ld bc, (scratch_0) ; HL BC = z_0 ** 2 - z_1 ** 2
ld c, b ; Divide by scale = 256
ld b, l ; Discard the rest
push bc ; We need BC later
- z3 = 2 * z0 * z1 / SCALE;
ld hl, (z_0) ; Compute DE HL = 2 * z_0 * z_1
add hl, hl
ld de, hl
ld bc, (z_1)
call mul_16
ld b, e ; Divide by scale (= 256)
ld c, h ; BC contains now z_3
- z1 = z3 + y;
ld hl, (y)
add hl, bc
ld (z_1), hl
- z_0 = z_2 + x;
pop bc ; Here BC is needed again :-)
ld hl, (x)
add hl, bc
ld (z_0), hl
- if (z0 * z0 / SCALE + z1 * z1 / SCALE > 4 * SCALE)
ld hl, (z_0_square_low) ; Use the squares computed
ld de, (z_1_square_low) ; above
add hl, de
ld bc, hl ; BC contains lower word of sum
ld hl, (z_0_square_high)
ld de, (z_1_square_high)
adc hl, de
ld h, l ; HL now contains (z_0 ** 2 +
ld l, b ; z_1 ** 2) / scale
ld bc, divergent
and a
sbc hl, bc
- break;
jp c, iteration_dec ; No break
pop bc ; Get latest iteration counter
jr iteration_end ; Exit loop
- iteration++;
iteration_dec pop bc ; Get iteration counter
djnz iteration_loop ; We might fall through!
- }
iteration_end
- printf("%c", display[iteration % 7]);
ld a, b
and $7 ; lower three bits only (c = 0)
sbc hl, hl
ld l, a
ld de, display ; Get start of character array
add hl, de ; address and load the
ld a, (hl) ; character to be printed
ld ix, _putc ; Print the character
rst 08
ld de, (x_step) ; x += x_step
ld hl, (x)
add hl, de
ld (x), hl
jp inner_loop
- }
- printf("\n");
inner_loop_end ld ix, _crlf ; Print a CR/LF pair
rst 08
ld de, (y_step) ; y += y_step
ld hl, (y)
add hl, de
ld (y), hl ; Store new y-value
jp outer_loop
- }
mandel_end ld hl, finished ; Print finished-message
ld ix, _puts
rst 08
rst 0 ; Return to the monitor
welcome defb "Generating a Mandelbrot set"
defb cr, lf, eos
finished defb "Computation finished.", cr, lf, eos
iteration_max defb 10 ; How many iterations x defw 0 ; x-coordinate x_start defw -2 * scale ; Minimum x-coordinate x_end defw 5 * scale / 10 ; Maximum x-coordinate x_step defw 4 * scale / 100 ; x-coordinate step-width y defw -1 * scale ; Minimum y-coordinate y_end defw 1 * scale ; Maximum y-coordinate y_step defw 1 * scale / 10 ; y-coordinate step-width z_0 defw 0 z_1 defw 0 scratch_0 defw 0 z_0_square_high defw 0 z_0_square_low defw 0 z_1_square_high defw 0 z_1_square_low defw 0 display defb " .-+*=#@" ; 8 characters for the display
- Compute DEHL = BC * DE (signed)
- This routine is not too clever but it
- works. It is based on a standard 16-by-16 multiplication routine for unsigned
- integers. At the beginning the sign of the result is determined based on the
- signs of the operands which are negated if necessary. Then the unsigned
- multiplication takes place, followed by negating the result if necessary.
mul_16 xor a ; Clear carry and A (-> +)
bit 7, b ; Is BC negative?
jr z, bc_positive ; No
sub c ; A is still zero, complement
ld c, a
ld a, 0
sbc a, b
ld b, a
scf ; Set carry (-> -)
bc_positive bit 7, D ; Is DE negative?
jr z, de_positive ; No
push af ; Remember carry for later!
xor a
sub e
ld e, a
ld a, 0
sbc a, d
ld d, a
pop af ; Restore carry for complement
ccf ; Complement Carry (-> +/-?)
de_positive push af ; Remember state of carry
and a ; Start multiplication
sbc hl, hl
ld a, 16 ; 16 rounds
mul_16_loop add hl, hl
rl e
rl d
jr nc, mul_16_exit
add hl, bc
jr nc, mul_16_exit
inc de
mul_16_exit dec a
jr nz, mul_16_loop
pop af ; Restore carry from beginning
ret nc ; No sign inversion necessary
xor a ; Complement DE HL
sub l
ld l, a
ld a, 0
sbc a, h
ld h, a
ld a, 0
sbc a, e
ld e, a
ld a, 0
sbc a, d
ld d, a
ret
</lang>
- Output:
Generating a Mandelbrot set ....... @@@@@@@@@@@@@@@@@@@@########===*+ . *######@@@@@ ...... @@@@@@@@@@@@@@@@@@@@#########====+-. -*===#####@@@@ ..... @@@@@@@@@@@@@@@@@@@@#########===**+- +*====####@@ .... @@@@@@@@@@@@@@@@@@@########==*+-.-+-.. .-+****+*##@ ... @@@@@@@@@@@@@@@@@#####=====**+- -=# .. @@@@@@@@@@@@@@@###=========***+- -*== ..@@@@@@@@@@####==- +*******++++- -+= . @@@@#######===**- ... -* .@@#######======++- += .#######=****+- +*= .=**++ .-.... .+*== .#=====****+++. -*== .@#######=====*--- .*= .@@@########====*++. += . @@@@@@@######==*+ .-++- --++--. = .. @@@@@@@@@@@@@##==*=======****+ -+*= ... @@@@@@@@@@@@@@@@####=======***+- +== ... @@@@@@@@@@@@@@@@@@@######====*- . --+- *## .... @@@@@@@@@@@@@@@@@@@@########==****++- .-+**====###@ ..... @@@@@@@@@@@@@@@@@@@@##########===*+ *===#####@@@ ....... @@@@@@@@@@@@@@@@@@@@@########===*+-. .+*=######@@@@@ Computation finished.
zkl
Uses the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl
<lang zkl>fcn mandelbrot{ // lord this is slooooow
bitmap:=PPM(640,480);
foreach y,x in ([0..479],[0..639]){
cx:=(x.toFloat()/640 - 0.5)*4.0; //range: -2.0 to +2.0
cy:=((y-240).toFloat()/240.0)*1.5; //range: -1.5 to +1.5
cnt:=0; zx:=0.0; zy:=0.0;
do(1000){
if(zx*zx + zy*zy > 2.0){ //z heads toward infinity
//set color of pixel to rate it approaches infinity bitmap[x,y]=cnt.shiftLeft(21) + cnt.shiftLeft(10) + cnt*8; break; } temp:=zx*zy; zx=zx*zx - zy*zy + cx; //calculate next iteration of z zy=2.0*temp + cy; cnt+=1;
}
}
bitmap.write(File("foo.ppm","wb"));
}();</lang>
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