Sierpinski triangle/Graphical
You are encouraged to solve this task according to the task description, using any language you may know.
Produce a graphical representation of a Sierpinski triangle of order N in any orientation.
An example of Sierpinski's triangle (order = 8) looks like this:
Asymptote
This simple-minded recursive apporach doesn't scale well to large orders, but neither would your PostScript viewer, so there's nothing to gain from a more efficient algorithm. Thus are the perils of vector graphics.
<lang asymptote>path subtriangle(path p, real node) {
return
point(p, node) --
point(p, node + 1/2) --
point(p, node - 1/2) --
cycle;
}
void sierpinski(path p, int order) {
if (order == 0)
fill(p);
else {
sierpinski(subtriangle(p, 0), order - 1);
sierpinski(subtriangle(p, 1), order - 1);
sierpinski(subtriangle(p, 2), order - 1);
}
}
sierpinski((0, 0) -- (5 inch, 1 inch) -- (2 inch, 6 inch) -- cycle, 10);</lang>
C
Code lifted from Dragon curve. Given a depth n, draws a triangle of size 2^n in a PNM file to the standard output. Usage: gcc -lm stuff.c -o sierp; ./sierp 9 > triangle.pnm. Sample image generated with depth 9. Generated image's size depends on the depth: it plots dots, but does not draw lines, so a large size with low depth is not possible.
<lang C>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <math.h>
long long x, y, dx, dy, scale, clen, cscale; typedef struct { double r, g, b; } rgb; rgb ** pix;
void sc_up() { scale *= 2; x *= 2; y *= 2; cscale *= 3; }
void h_rgb(long long x, long long y) { rgb *p = &pix[y][x];
- define SAT 1
double h = 6.0 * clen / cscale; double VAL = 1; double c = SAT * VAL; double X = c * (1 - fabs(fmod(h, 2) - 1));
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 iter_string(char * str, int d) { long long len; while (*str != '\0') { switch(*(str++)) { case 'X': if (d) iter_string("XHXVX", d - 1); else{ clen ++; h_rgb(x/scale, y/scale); x += dx; y -= dy; } continue; case 'V': len = 1LLU << d; while (len--) { clen ++; h_rgb(x/scale, y/scale); y += dy; } continue; case 'H': len = 1LLU << d; while(len --) { clen ++; h_rgb(x/scale, y/scale); x -= dx; } continue; } } }
void sierp(long leng, int depth) { long i; long h = leng + 20, w = leng + 20;
/* allocate pixel buffer */ rgb *buf = malloc(sizeof(rgb) * w * h); pix = malloc(sizeof(rgb *) * h); for (i = 0; i < h; i++) pix[i] = buf + w * i; memset(buf, 0, sizeof(rgb) * w * h);
/* init coords; scale up to desired; exec string */
x = y = 10; dx = leng; dy = leng; scale = 1; clen = 0; cscale = 3; for (i = 0; i < depth; i++) sc_up(); iter_string("VXH", depth);
/* write color PNM file */ unsigned char *fpix = malloc(w * h * 3); double maxv = 0, *dbuf = (double*)buf;
for (i = 3 * w * h - 1; i >= 0; i--) if (dbuf[i] > maxv) maxv = dbuf[i]; for (i = 3 * h * w - 1; i >= 0; i--) fpix[i] = 255 * dbuf[i] / maxv;
printf("P6\n%ld %ld\n255\n", w, h); fflush(stdout); /* printf and fwrite may treat buffer differently */ fwrite(fpix, h * w * 3, 1, stdout); }
int main(int c, char ** v) { int size, depth;
depth = (c > 1) ? atoi(v[1]) : 10; size = 1 << depth;
fprintf(stderr, "size: %d depth: %d\n", size, depth); sierp(size, depth + 2);
return 0; }</lang>
Go
<lang go>package main
import (
"fmt" "image" "image/color" "image/draw" "image/png" "os"
)
func main() {
const order = 8
const width = 1 << order
const margin = 10
bounds := image.Rect(-margin, -margin, width+2*margin, width+2*margin)
im := image.NewGray(bounds)
gBlack := color.Gray{0}
gWhite := color.Gray{255}
draw.Draw(im, bounds, image.NewUniform(gWhite), image.ZP, draw.Src)
for y := 0; y < width; y++ {
for x := 0; x < width; x++ {
if x&y == 0 {
im.SetGray(x, y, gBlack)
}
}
}
f, err := os.Create("sierpinski.png")
if err != nil {
fmt.Println(err)
return
}
if err = png.Encode(f, im); err != nil {
fmt.Println(err)
}
if err = f.Close(); err != nil {
fmt.Println(err)
}
}</lang>
Haskell
This program uses the diagrams package to produce the Sierpinski triangle. The package implements an embedded DSL for producing vector graphics. Depending on the command-line arguments, the program can generate SVG, PNG, PDF or PostScript output.
For fun, we take advantage of Haskell's layout rules, and the operators provided by the diagrams package, to give the reduce function the shape of a triangle. It could also be written as reduce t = t === (t ||| t).
The command to produce the SVG output is sierpinski -o Sierpinski-Haskell.svg.
<lang haskell>import Diagrams.Prelude import Diagrams.Backend.Cairo.CmdLine
triangle = eqTriangle # fc black # lw 0
reduce t = t
===
(t ||| t)
sierpinski = iterate reduce triangle
main = defaultMain $ sierpinski !! 7 </lang>
Icon and Unicon
The following code is adapted from a program by Ralph Griswold that demonstrates an interesting way to draw the Sierpinski Triangle. Given an argument of the order it will calculate the canvas size needed with margin. It will not stop you from asking for a triangle larger than you display. For an explanation, see "Chaos and Fractals", Heinz-Otto Peitgen, Harmut Jurgens, and Dietmar Saupe, Springer-Verlah, 1992, pp. 132-134.
<lang Icon>link wopen
procedure main(A)
local width, margin, x, y
width := 2 ^ (order := (0 < integer(\A[1])) | 8)
wsize := width + 2 * (margin := 30 )
WOpen("label=Sierpinski", "size="||wsize||","||wsize) |
stop("*** cannot open window")
every y := 0 to width - 1 do
every x := 0 to width - 1 do
if iand(x, y) = 0 then DrawPoint(x + margin, y + margin)
Event()
end</lang>
Original source IPL Graphics/sier1.
J
Solution: <lang j> load 'viewmat'
'rgb'viewmat--. |. (~:_1&|.)^:(<@#) (2^8){.1
</lang>
or
<lang j> load'viewmat' viewmat(,~,.~)^:8,1 </lang>
Liberty BASIC
The ability of LB to handle very large integers makes the Pascal triangle method very attractive. If you alter the rem'd line you can ask it to print the last, central term... <lang lb> nomainwin
open "test" for graphics_nsb_fs as #gr
- gr "trapclose quit"
- gr "down; home"
- gr "posxy cx cy"
order =10
w =cx *2: h =cy *2
dim a( h, h) 'line, col
- gr "trapclose quit"
- gr "down; home"
a( 1, 1) =1
for i = 2 to 2^order -1
scan
a( i, 1) =1
a( i, i) =1
for j = 2 to i -1
'a(i,j)=a(i-1,j-1)+a(i-1,j) 'LB is quite capable for crunching BIG numbers
a( i, j) =(a( i -1, j -1) +a( i -1, j)) mod 2 'but for this task, last bit is enough (and it much faster)
next
for j = 1 to i
if a( i, j) mod 2 then #gr "set "; cx +j -i /2; " "; i
next
next
- gr "flush"
wait
sub quit handle$
close #handle$ end
end sub </lang> Up to order 10 displays on a 1080 vertical pixel screen.
Logo
This will draw a graphical Sierpinski gasket using turtle graphics. <lang logo>to sierpinski :n :length
if :n = 0 [stop] repeat 3 [sierpinski :n-1 :length/2 fd :length rt 120]
end seth 30 sierpinski 5 200</lang>
Mathematica
<lang Mathematica>Sierpinski[n_] :=Nest[Flatten[Table[{{
#i, 1, (#i, 1 + #i, 2)/2, (#i, 1 + #i, 3)/ 2}, {(#i, 1 + #i, 2)/2, #[[i, 2]], (#i, 2 + #i, 3)/2}, {(#i, 1 + #i, 3)/ 2, (#i, 2 + #i, 3)/2, #i, 3}}, {i, Length[#]}], 1] &, {{{0, 0}, {1/2, 1}, {1, 0}}}, n]
Show[Graphics[{Opacity[1], Black, Map[Polygon, Sierpinski[8], 1]}, AspectRatio -> 1]]</lang>
OCaml
<lang ocaml>open Graphics
let round v =
int_of_float (floor (v +. 0.5))
let middle (x1, y1) (x2, y2) =
((x1 +. x2) /. 2.0, (y1 +. y2) /. 2.0)
let draw_line (x1, y1) (x2, y2) =
moveto (round x1) (round y1); lineto (round x2) (round y2);
let draw_triangle (p1, p2, p3) =
draw_line p1 p2; draw_line p2 p3; draw_line p3 p1;
let () =
open_graph "";
let width = float (size_x ()) in
let height = float (size_y ()) in
let pad = 20.0 in
let initial_triangle =
( (pad, pad),
(width -. pad, pad),
(width /. 2.0, height -. pad) )
in
let rec loop step tris =
if step <= 0 then tris else
loop (pred step) (
List.fold_left (fun acc (p1, p2, p3) ->
let m1 = middle p1 p2
and m2 = middle p2 p3
and m3 = middle p3 p1 in
let tri1 = (p1, m1, m3)
and tri2 = (p2, m2, m1)
and tri3 = (p3, m3, m2) in
tri1 :: tri2 :: tri3 :: acc
) [] tris
)
in
let res = loop 6 [ initial_triangle ] in
List.iter draw_triangle res;
ignore (read_key ())</lang>
run with:
ocaml graphics.cma sierpinski.ml
Perl
Writes out an EPS given an arbitrary triangle. The perl code only calculates the bounding box, while real work is done in postscript. <lang Perl>use List::Util qw'min max sum';
sub write_eps { my @x = @_[0, 2, 4]; my @y = @_[1, 3, 5]; my $sx = sum(@x) / 3; my $sy = sum(@y) / 3; @x = map { $_ - $sx } @x; @y = map { $_ - $sy } @y;
print <<"HEAD"; %!PS-Adobe-3.0 %%BoundingBox: @{[min(@x) - 10]} @{[min(@y) - 10]} @{[max(@x) + 10]} @{[max(@y) + 10]} /v1 { $x[0] $y[0] } def /v2 { $x[1] $y[1] } def /v3 { $x[2] $y[2] } def /t { translate } def /r { .5 .5 scale 2 copy t 2 index sierp pop neg exch neg exch t 2 2 scale } def
/sierp { dup 1 sub dup 0 ne { v1 r v2 r v3 r } { v1 moveto v2 lineto v3 lineto} ifelse pop } def
9 sierp fill pop showpage %%EOF HEAD }
write_eps 0, 0, 300, 215, -25, 200;</lang>
Perl 6
This is a recursive solution. It is not really practical for more than 8 levels of recursion, but anything more than 7 is barely visible anyway. <lang perl6>my $side = 512; my $height = get_height($side); my $levels = 8;
sub get_height ($side) { $side * 3.sqrt / 2 }
sub triangle ( $x1, $y1, $x2, $y2, $x3, $y3, $fill?, $animate? ) {
print "<polygon points=\"$x1,$y1 $x2,$y2 $x3,$y3\"";
if $fill { print " style=\"fill: $fill; stroke-width: 0;\"" };
if $animate
{
say ">\n <animate attributeType=\"CSS\" attributeName=\"opacity\"\n values=\"1;0;1\""
~ " keyTimes=\"0;.5;1\" dur=\"20s\" repeatCount=\"indefinite\" />\n</polygon>"
}
else
{
say ' />';
}
}
sub fractal ( $x1, $y1, $x2, $y2, $x3, $y3, $r is copy ) {
triangle( $x1, $y1, $x2, $y2, $x3, $y3 );
return unless --$r;
my $side = abs($x3 - $x2) / 2;
my $height = get_height($side);
fractal( $x1, $y1-$height*2, $x1-$side/2, $y1-3*$height, $x1+$side/2, $y1-3*$height, $r);
fractal( $x2, $y1, $x2-$side/2, $y1-$height, $x2+$side/2, $y1-$height, $r);
fractal( $x3, $y1, $x3-$side/2, $y1-$height, $x3+$side/2, $y1-$height, $r);
}
say '<?xml version="1.0" standalone="no"?> <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"> <svg width="100%" height="100%" version="1.1" xmlns="http://www.w3.org/2000/svg"> <defs>
<radialGradient id="basegradient" cx="50%" cy="65%" r="50%" fx="50%" fy="65%"> <stop offset="10%" stop-color="#ff0" /> <stop offset="60%" stop-color="#f00" /> <stop offset="99%" stop-color="#00f" /> </radialGradient>
</defs>';
triangle( $side/2, 0, 0, $height, $side, $height, 'url(#basegradient)' ); triangle( $side/2, 0, 0, $height, $side, $height, '#000', 'animate' ); say '<g style="fill: #fff; stroke-width: 0;">'; fractal( $side/2, $height, $side*3/4, $height/2, $side/4, $height/2, $levels ); say '</g></svg>';</lang>
PicoLisp
Slight modification of the text version, requires ImageMagick's display: <lang PicoLisp>(de sierpinski (N)
(let (D '("1") S "0")
(do N
(setq
D (conc
(mapcar '((X) (pack S X S)) D)
(mapcar '((X) (pack X "0" X)) D) )
S (pack S S) ) )
D ) )
(out '(display -)
(let Img (sierpinski 7)
(prinl "P1")
(prinl (length (car Img)) " " (length Img))
(mapc prinl Img) ) )
</lang>
PostScript
<lang PostScript>%!PS
/sierp { % level ax ay bx by cx cy
6 cpy triangle sierpr
} bind def
/sierpr {
12 cpy
10 -4 2 {
5 1 roll exch 4 -1 roll
add 0.5 mul 3 1 roll
add 0.5 mul 3 -1 roll
2 roll
} for % l a b c bc ac ab
13 -1 roll dup 0 gt {
1 sub
dup 4 cpy 18 -2 roll sierpr
dup 7 index 7 index 2 cpy 16 -2 roll sierpr
9 3 roll 1 index 1 index 2 cpy 13 4 roll sierpr
} { 13 -6 roll 7 { pop } repeat } ifelse
triangle
} bind def
/cpy { { 5 index } repeat } bind def
/triangle {
newpath moveto lineto lineto closepath stroke
} bind def
6 50 100 550 100 300 533 sierp showpage</lang>
Prolog
Works with SWI-Prolog and XPCE.
Recursive version
Works up to sierpinski(13). <lang Prolog>sierpinski(N) :- sformat(A, 'Sierpinski order ~w', [N]), new(D, picture(A)), draw_Sierpinski(D, N, point(350,50), 600), send(D, size, size(690,690)), send(D, open).
draw_Sierpinski(Window, 1, point(X, Y), Len) :- X1 is X - round(Len/2), X2 is X + round(Len/2), Y1 is Y + Len * sqrt(3) / 2, send(Window, display, new(Pa, path)),
(
send(Pa, append, point(X, Y)), send(Pa, append, point(X1, Y1)), send(Pa, append, point(X2, Y1)), send(Pa, closed, @on), send(Pa, fill_pattern, colour(@default, 0, 0, 0)) ).
draw_Sierpinski(Window, N, point(X, Y), Len) :-
Len1 is round(Len/2),
X1 is X - round(Len/4),
X2 is X + round(Len/4),
Y1 is Y + Len * sqrt(3) / 4,
N1 is N - 1,
draw_Sierpinski(Window, N1, point(X, Y), Len1),
draw_Sierpinski(Window, N1, point(X1, Y1), Len1),
draw_Sierpinski(Window, N1, point(X2, Y1), Len1).</lang>
Iterative version
<lang Prolog>:- dynamic top/1.
sierpinski_iterate(N) :- retractall(top(_)), sformat(A, 'Sierpinski order ~w', [N]), new(D, picture(A)), draw_Sierpinski_iterate(D, N, point(550, 50)), send(D, open).
draw_Sierpinski_iterate(Window, N, point(X,Y)) :- assert(top([point(X,Y)])), NbTours is 2 ** (N - 1), % Size is given here to preserve the "small" triangles when N is big Len is 10, forall(between(1, NbTours, _I), ( retract(top(Lst)), assert(top([])), forall(member(P, Lst), draw_Sierpinski(Window, P, Len)))).
draw_Sierpinski(Window, point(X, Y), Len) :- X1 is X - round(Len/2), X2 is X + round(Len/2), Y1 is Y + round(Len * sqrt(3) / 2), send(Window, display, new(Pa, path)),
(
send(Pa, append, point(X, Y)), send(Pa, append, point(X1, Y1)), send(Pa, append, point(X2, Y1)), send(Pa, closed, @on), send(Pa, fill_pattern, colour(@default, 0, 0, 0)) ), retract(top(Lst)), ( member(point(X1, Y1), Lst) -> select(point(X1,Y1), Lst, Lst1) ; Lst1 = [point(X1, Y1)|Lst]),
( member(point(X2, Y1), Lst1) -> select(point(X2,Y1), Lst1, Lst2) ; Lst2 = [point(X2, Y1)|Lst1]),
assert(top(Lst2)).</lang>
Python
<lang python>
- !/usr/bin/env python
- import necessary modules
- ------------------------
from numpy import * import turtle
- Functions defining the drawing actions (used by the function DrawSierpinskiTriangle).
- -------------------------------------------------------------------------------------
def Left(turn, point, fwd, angle, turt): turt.left(angle) return [turn, point, fwd, angle, turt] def Right(turn, point, fwd, angle, turt): turt.right(angle) return [turn, point, fwd, angle, turt] def Forward(turn, point, fwd, angle, turt): turt.forward(fwd) return [turn, point, fwd, angle, turt]
- The drawing function
- --------------------
- level level of Sierpinski triangle (minimum value = 1)
- ss screensize (Draws on a screen of size ss x ss. Default value = 400.)
- -----------------------------------------------------------------------------------------------
def DrawSierpinskiTriangle(level, ss=400): # typical values turn = 0 # initial turn (0 to start horizontally) angle=60.0 # in degrees
# Initialize the turtle turtle.hideturtle() turtle.screensize(ss,ss) turtle.penup() turtle.degrees()
# The starting point on the canvas fwd0 = float(ss) point=array([-fwd0/2.0, -fwd0/2.0])
# Setting up the Lindenmayer system # Assuming that the triangle will be drawn in the following way: # 1.) Start at a point # 2.) Draw a straight line - the horizontal line (H) # 3.) Bend twice by 60 degrees to the left (--) # 4.) Draw a straight line - the slanted line (X) # 5.) Bend twice by 60 degrees to the left (--) # 6.) Draw a straight line - another slanted line (X) # This produces the triangle in the first level. (so the axiom to begin with is H--X--X) # 7.) For the next level replace each horizontal line using # X->XX # H -> H--X++H++X--H # The lengths will be halved.
decode = {'-':Left, '+':Right, 'X':Forward, 'H':Forward}
axiom = 'H--X--X'
# Start the drawing turtle.goto(point[0], point[1]) turtle.pendown() turtle.hideturtle() turt=turtle.getpen() startposition=turt.clone()
# Get the triangle in the Lindenmayer system fwd = fwd0/(2.0**level) path = axiom for i in range(0,level): path=path.replace('X','XX') path=path.replace('H','H--X++H++X--H')
# Draw it. for i in path: [turn, point, fwd, angle, turt]=decode[i](turn, point, fwd, angle, turt)
DrawSierpinskiTriangle(5)
</lang>
Ruby
<lang ruby>Shoes.app(:height=>540,:width=>540, :title=>"Sierpinski Triangle") do
def triangle(slot, tri, color)
x, y, len = tri
slot.append do
fill color
shape do
move_to(x,y)
dx = len * Math::cos(Math::PI/3)
dy = len * Math::sin(Math::PI/3)
line_to(x-dx, y+dy)
line_to(x+dx, y+dy)
line_to(x,y)
end
end
end
@s = stack(:width => 520, :height => 520) {}
@s.move(10,10)
length = 512 @triangles = length/2,0,length triangle(@s, @triangles[0], rgb(0,0,0))
@n = 1
animate(1) do
if @n <= 7
@triangles = @triangles.inject([]) do |sum, (x, y, len)|
dx = len/2 * Math::cos(Math::PI/3)
dy = len/2 * Math::sin(Math::PI/3)
triangle(@s, [x, y+2*dy, -len/2], rgb(255,255,255))
sum += [[x, y, len/2], [x-dx, y+dy, len/2], [x+dx, y+dy, len/2]]
end
end
@n += 1
end
keypress do |key| case key when :control_q, "\x11" then exit end end
end</lang>
Run BASIC
<lang runbasic>graphic #g, 300,300 order = 8 width = 100 w = width * 11 dim canvas(w,w) canvas(1,1) = 1
for x = 2 to 2^order -1
canvas(x,1) = 1
canvas(x,x) = 1
for y = 2 to x -1
canvas( x, y) = (canvas(x -1,y -1) + canvas(x -1, y)) mod 2
if canvas(x,y) mod 2 then #g "set "; width + (order*3) + y - x / 2;" "; x
next y
next x render #g
- g "flush"
wait</lang>
Tcl
This code maintains a queue of triangles to cut out; though a stack works just as well, the observed progress is more visually pleasing when a queue is used.
<lang tcl>package require Tcl 8.5 package require Tk
proc mean args {expr {[::tcl::mathop::+ {*}$args] / [llength $args]}} proc sierpinski {canv coords order} {
$canv create poly $coords -fill black -outline {}
set queue [list [list {*}$coords $order]]
while {[llength $queue]} {
lassign [lindex $queue 0] x1 y1 x2 y2 x3 y3 order set queue [lrange $queue 1 end] if {[incr order -1] < 0} continue set x12 [mean $x1 $x2]; set y12 [mean $y1 $y2] set x23 [mean $x2 $x3]; set y23 [mean $y2 $y3] set x31 [mean $x3 $x1]; set y31 [mean $y3 $y1] $canv create poly $x12 $y12 $x23 $y23 $x31 $y31 -fill white -outline {} update idletasks; # So we can see progress lappend queue [list $x1 $y1 $x12 $y12 $x31 $y31 $order] \ [list $x12 $y12 $x2 $y2 $x23 $y23 $order] \ [list $x31 $y31 $x23 $y23 $x3 $y3 $order]
}
}
pack [canvas .c -width 400 -height 400 -background white] update; # So we can see progress sierpinski .c {200 10 390 390 10 390} 7</lang>