Sierpinski pentagon: Difference between revisions
| Line 576: | Line 576: | ||
varr = [vertices[parse(Int, ch) + 1] for ch in split(string(i, base=sides, pad=order), "")] |
varr = [vertices[parse(Int, ch) + 1] for ch in split(string(i, base=sides, pad=order), "")] |
||
vector = sum(map(x -> varr[x] * orders[x], 1:length(orders))) |
vector = sum(map(x -> varr[x] * orders[x], 1:length(orders))) |
||
vprod = map(x->vector+orders[end]*(1-scale)*x, vertices) |
vprod = map(x -> vector + orders[end] * (1-scale) * x, vertices) |
||
points = join([@sprintf("%.3f %.3f", real(v), imag(v)) for v in vprod], " ") |
points = join([@sprintf("%.3f %.3f", real(v), imag(v)) for v in vprod], " ") |
||
Revision as of 22:53, 5 January 2019
You are encouraged to solve this task according to the task description, using any language you may know.
Produce a graphical or ASCII-art representation of a Sierpinski pentagon (aka a Pentaflake) of order 5. Your code should also be able to correctly generate representations of lower orders: 1 to 4.
- See also
C
The Sierpinski fractals can be generated via the Chaos Game. This implementation thus generalizes the Chaos game C implementation on Rosettacode. As the number of sides increases, the number of iterations must increase dramatically for a well pronounced fractal ( 30000 for a pentagon). This is in keeping with the requirements that the implementation should work for polygons with sides 1 to 4 as well. Requires the WinBGIm library. <lang C>
- include<graphics.h>
- include<stdlib.h>
- include<stdio.h>
- include<math.h>
- include<time.h>
- define pi M_PI
int main(){
time_t t; double side, **vertices,seedX,seedY,windowSide = 500,sumX=0,sumY=0; int i,iter,choice,numSides;
printf("Enter number of sides : "); scanf("%d",&numSides);
printf("Enter polygon side length : "); scanf("%lf",&side);
printf("Enter number of iterations : "); scanf("%d",&iter);
initwindow(windowSide,windowSide,"Polygon Chaos");
vertices = (double**)malloc(numSides*sizeof(double*));
for(i=0;i<numSides;i++){ vertices[i] = (double*)malloc(2 * sizeof(double));
vertices[i][0] = windowSide/2 + side*cos(i*2*pi/numSides); vertices[i][1] = windowSide/2 + side*sin(i*2*pi/numSides); sumX+= vertices[i][0]; sumY+= vertices[i][1]; putpixel(vertices[i][0],vertices[i][1],15); }
srand((unsigned)time(&t));
seedX = sumX/numSides; seedY = sumY/numSides;
putpixel(seedX,seedY,15);
for(i=0;i<iter;i++){ choice = rand()%numSides;
seedX = (seedX + (numSides-2)*vertices[choice][0])/(numSides-1); seedY = (seedY + (numSides-2)*vertices[choice][1])/(numSides-1);
putpixel(seedX,seedY,15); }
free(vertices);
getch();
closegraph();
return 0; } </lang>
D
This solution combines the turtle graphics concept used in Python, with the SVG output format of the Perl 6 solution. This runs very quickly compared to the Python version.
<lang D>import std.math; import std.stdio;
/// Convert degrees into radians, as that is the accepted unit for sin/cos etc... real degrees(real deg) {
immutable tau = 2.0 * PI; return deg * tau / 360.0;
}
immutable part_ratio = 2.0 * cos(72.degrees); immutable side_ratio = 1.0 / (part_ratio + 2.0);
/// Use the provided turtle to draw a pentagon of the specified size void pentagon(Turtle turtle, real size) {
turtle.right(36.degrees);
turtle.begin_fill();
foreach(i; 0..5) {
turtle.forward(size);
turtle.right(72.degrees);
}
turtle.end_fill();
}
/// Draw a sierpinski pentagon of the desired order void sierpinski(int order, Turtle turtle, real size) {
turtle.setheading(0.0); auto new_size = size * side_ratio;
if (order-- > 1) {
// create four more turtles
foreach(j; 0..4) {
turtle.right(36.degrees);
real small = size * side_ratio / part_ratio;
auto dist = [small, size, size, small][j];
auto spawn = new Turtle();
spawn.setposition(turtle.position);
spawn.setheading(turtle.heading);
spawn.forward(dist);
// recurse for each spawned turtle
sierpinski(order, spawn, new_size);
}
// recurse for the original turtle
sierpinski(order, turtle, new_size);
} else {
// The bottom has been reached for this turtle
pentagon(turtle, size);
}
}
/// Run the generation of a P(5) sierpinksi pentagon void main() {
int order = 5; real size = 500;
auto turtle = new Turtle(size/2, size);
// Write the header to an SVG file for the image writeln(`<?xml version="1.0" standalone="no"?>`); writeln(`<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"`); writeln(` "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">`); writefln(`<svg height="%s" width="%s" style="fill:blue" transform="translate(%s,%s) rotate(-36)"`, size, size, size/2, size/2); writeln(` version="1.1" xmlns="http://www.w3.org/2000/svg">`); // Write the close tag when the interior points have been written scope(success) writeln("</svg>");
// Scale the initial turtle so that it stays in the inner pentagon size *= part_ratio;
// Begin rendering sierpinski(order, turtle, size);
}
/// Define a position struct Point {
real x; real y;
/// When a point is written, do it in the form "x,y " to three decimal places
void toString(scope void delegate(const(char)[]) sink) const {
import std.format;
formattedWrite(sink, "%0.3f", x);
sink(",");
formattedWrite(sink, "%0.3f", y);
sink(" ");
}
}
/// Mock turtle implementation sufficiant to handle "drawing" the pentagons class Turtle {
///////////////////////////////// private:
Point pos; real theta; bool tracing;
/////////////////////////////////
public:
this() {
// empty
}
this(real x, real y) {
pos.x = x;
pos.y = y;
}
// Get/Set the turtle position
Point position() {
return pos;
}
void setposition(Point pos) {
this.pos = pos;
}
// Get/Set the turtle's heading
real heading() {
return theta;
}
void setheading(real angle) {
theta = angle;
}
// Move the turtle through space
void forward(real dist) {
// Calculate both components at once for the specified angle
auto delta = dist * expi(theta);
pos.x += delta.re;
pos.y += delta.im;
if (tracing) {
write(pos);
}
}
// Turn the turle
void right(real angle) {
theta = theta - angle;
}
// Start/Stop exporting the points of the polygon
void begin_fill() {
write(`<polygon points="`);
tracing = true;
}
void end_fill() {
writeln(`"/>`);
tracing = false;
}
}</lang>
Go
This follows the approach of the Java entry but uses a fixed palette of 5 colors which are selected in order rather than randomly.
As output is to an external .png file, only a pentaflake of order 5 is drawn though pentaflakes of lower orders can still be drawn by setting the 'order' variable to the appropriate figure. <lang go>package main
import (
"github.com/fogleman/gg" "image/color" "math"
)
var (
red = color.RGBA{255, 0, 0, 255}
green = color.RGBA{0, 255, 0, 255}
blue = color.RGBA{0, 0, 255, 255}
magenta = color.RGBA{255, 0, 255, 255}
cyan = color.RGBA{0, 255, 255, 255}
)
var (
w, h = 640, 640
dc = gg.NewContext(w, h)
deg72 = gg.Radians(72)
scaleFactor = 1 / (2 + math.Cos(deg72)*2)
palette = [5]color.Color{red, green, blue, magenta, cyan}
colorIndex = 0
)
func drawPentagon(x, y, side float64, depth int) {
angle := 3 * deg72
if depth == 0 {
dc.MoveTo(x, y)
for i := 0; i < 5; i++ {
x += math.Cos(angle) * side
y -= math.Sin(angle) * side
dc.LineTo(x, y)
angle += deg72
}
dc.SetColor(palette[colorIndex])
dc.Fill()
colorIndex = (colorIndex + 1) % 5
} else {
side *= scaleFactor
dist := side * (1 + math.Cos(deg72)*2)
for i := 0; i < 5; i++ {
x += math.Cos(angle) * dist
y -= math.Sin(angle) * dist
drawPentagon(x, y, side, depth-1)
angle += deg72
}
}
}
func main() {
dc.SetRGB(1, 1, 1) // White background
dc.Clear()
order := 5 // Can also set this to 1, 2, 3 or 4
hw := float64(w / 2)
margin := 20.0
radius := hw - 2*margin
side := radius * math.Sin(math.Pi/5) * 2
drawPentagon(hw, 3*margin, side, order-1)
dc.SavePNG("sierpinski_pentagon.png")
}</lang>
- Output:
Image similar to Java entry but uses a fixed palette of colors.
Haskell
For universal solution see Fractal tree#Haskell
<lang haskell>import Graphics.Gloss
pentaflake :: Int -> Picture pentaflake order = iterate transformation pentagon !! order
where transformation = Scale s s . foldMap copy [0,72..288] copy a = Rotate a . Translate 0 x pentagon = Polygon [ (sin a, cos a) | a <- [0,2*pi/5..2*pi] ] x = 2*cos(pi/5) s = 1/(1+x)
main = display dc white (Color blue $ Scale 300 300 $ pentaflake 5)
where dc = InWindow "Pentaflake" (400, 400) (0, 0)</lang>
Explanation: Since Picture forms a monoid with image overlaying as multiplication, so do functions having type Picture -> Picture:
f,g :: Picture -> Picture f <> g = \p -> f p <> g p
Function copy for an angle returns transformation, which shifts and rotates given picture, therefore foldMap copy for a list of angles returns a transformation, which shifts and rotates initial image five times. After that the resulting image is scaled to fit the inital size, so that it is ready for next iteration.
If one wants to get all intermediate pentaflakes transformation shoud be changed as follows:
<lang haskell>transformation = Scale s s . (Rotate 36 <> foldMap copy [0,72..288])</lang>
See also the implementation using Diagrams
Java
<lang java>import java.awt.*; import java.awt.event.ActionEvent; import java.awt.geom.Path2D; import static java.lang.Math.*; import java.util.Random; import javax.swing.*;
public class SierpinskiPentagon extends JPanel {
// exterior angle final double degrees072 = toRadians(72);
/* After scaling we'll have 2 sides plus a gap occupying the length
of a side before scaling. The gap is the base of an isosceles triangle
with a base angle of 72 degrees. */
final double scaleFactor = 1 / (2 + cos(degrees072) * 2);
final int margin = 20; int limit = 0; Random r = new Random();
public SierpinskiPentagon() {
setPreferredSize(new Dimension(640, 640));
setBackground(Color.white);
new Timer(3000, (ActionEvent e) -> {
limit++;
if (limit >= 5)
limit = 0;
repaint();
}).start();
}
void drawPentagon(Graphics2D g, double x, double y, double side, int depth) {
double angle = 3 * degrees072; // starting angle
if (depth == 0) {
Path2D p = new Path2D.Double();
p.moveTo(x, y);
// draw from the top
for (int i = 0; i < 5; i++) {
x = x + cos(angle) * side;
y = y - sin(angle) * side;
p.lineTo(x, y);
angle += degrees072;
}
g.setColor(RandomHue.next());
g.fill(p);
} else {
side *= scaleFactor;
/* Starting at the top of the highest pentagon, calculate
the top vertices of the other pentagons by taking the
length of the scaled side plus the length of the gap. */
double distance = side + side * cos(degrees072) * 2;
/* The top positions form a virtual pentagon of their own,
so simply move from one to the other by changing direction. */
for (int i = 0; i < 5; i++) {
x = x + cos(angle) * distance;
y = y - sin(angle) * distance;
drawPentagon(g, x, y, side, depth - 1);
angle += degrees072;
}
}
}
@Override
public void paintComponent(Graphics gg) {
super.paintComponent(gg);
Graphics2D g = (Graphics2D) gg;
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON);
int w = getWidth();
double radius = w / 2 - 2 * margin;
double side = radius * sin(PI / 5) * 2;
drawPentagon(g, w / 2, 3 * margin, side, limit); }
public static void main(String[] args) {
SwingUtilities.invokeLater(() -> {
JFrame f = new JFrame();
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.setTitle("Sierpinski Pentagon");
f.setResizable(true);
f.add(new SierpinskiPentagon(), BorderLayout.CENTER);
f.pack();
f.setLocationRelativeTo(null);
f.setVisible(true);
});
}
}
class RandomHue {
/* Try to avoid random color values clumping together */ final static double goldenRatioConjugate = (sqrt(5) - 1) / 2; private static double hue = Math.random();
static Color next() {
hue = (hue + goldenRatioConjugate) % 1;
return Color.getHSBColor((float) hue, 1, 1);
}
}</lang>
JavaScript
- Notes
- I didn't try to, but got the first of 2 possible versions according to WP N-flake article. Mine has central pentagon. All others here got second version.
- This one looks a little bit differently from the 1st version on WP. Almost like 2nd version, but with central pentagon.
- Not a Durer's pentagon either.
<lang html>
<html>
<head>
<script type="application/x-javascript">
// Globals
var cvs, ctx, scale=500, p0, ord=0, clr='blue', jc=0;
var clrs=['blue','navy','green','darkgreen','red','brown','yellow','cyan'];
function p5f() {
cvs = document.getElementById("cvsid");
ctx = cvs.getContext("2d");
cvs.onclick=iter;
pInit(); //init plot
}
function iter() {
if(ord>5) {resetf(0)};
ctx.clearRect(0,0,cvs.width,cvs.height);
p0.forEach(iter5);
p0.forEach(pIter5);
ord++; document.getElementById("p1id").innerHTML=ord;
}
function iter5(v, i, a) {
if(typeof(v[0][0]) == "object") {a[i].forEach(iter5)}
else {a[i] = meta5(v)}
}
function pIter5(v, i, a) {
if(typeof(v[0][0]) == "object") {v.forEach(pIter5)}
else {pPoly(v)}
}
function pInit() {
p0 = [make5([.5,.5], .5)]; pPoly(p0[0]);
}
function meta5(h) {
c=h[0]; p1=c; p2=h[1]; z1=p1[0]-p2[0]; z2=p1[1]-p2[1];
dist = Math.sqrt(z1*z1 + z2*z2)/2.65;
nP=[];
for(k=1; k<h.length; k++) {
p1=h[k]; p2=c; a=Math.atan2(p2[1]-p1[1], p2[0]-p1[0]);
nP[k] = make5(ppad(a, dist, h[k]), dist)
}
nP[0]=make5(c, dist);
return nP;
}
function make5(c, r) {
vs=[]; j = 1;
for(i=1/10; i<2; i+=2/5) {
vs[j]=ppad(i*Math.PI, r, c); j++;
}
vs[0] = c; return vs;
}
function pPoly(s) {
ctx.beginPath(); ctx.moveTo(s[1][0]*scale, s[1][1]*-scale+scale); for(i=2; i<s.length; i++) ctx.lineTo(s[i][0]*scale, s[i][1]*-scale+scale); ctx.fillStyle=clr; ctx.fill()
}
// a - angle, d - distance, p - point function ppad(a, d, p) {
x=p[0]; y=p[1]; x2=d*Math.cos(a)+x; y2=d*Math.sin(a)+y; return [x2,y2]
}
function resetf(rord) {
ctx.clearRect(0,0,cvs.width,cvs.height);
ord=rord; jc++; if(jc>7){jc=0}; clr=clrs[jc];
document.getElementById("p1id").innerHTML=ord;
p5f();
} </script> </head>
<body onload="p5f()" style="font-family: arial, helvatica, sans-serif;"> Click Pentaflake to iterate. Order: <label id='p1id'>0</label> <input type="submit" value="RESET" onclick="resetf(0);"> (Reset anytime: to start new Pentaflake and change color.)
<canvas id="cvsid" width=640 height=640></canvas> </body>
</html> </lang>
- Output:
Page with Pentaflakejs.png Clicking Pentaflake you can see orders 1-6 of it in different colors.
Julia
<lang julia>using Printf
const sides = 5 const order = 5 const dim = 250 const scale = (3 - order ^ 0.5) / 2 const τ = 8 * atan(1, 1) const orders = map(x -> ((1 - scale) * dim) * scale ^ x, 0:order-1) cis(x) = Complex(cos(x), sin(x)) vertices = map(x -> cis(x * τ / sides), 0:sides-1)
fh = open("sierpinski_pentagon.svg", "w") print(fh, """<svg height=\"$(dim*2)\" width=\"$(dim*2)\" style=\"fill:blue\" """ *
"""version=\"1.1\" xmlns=\"http://www.w3.org/2000/svg\">\n""")
for i in 1:sides^order
varr = [vertices[parse(Int, ch) + 1] for ch in split(string(i, base=sides, pad=order), "")] vector = sum(map(x -> varr[x] * orders[x], 1:length(orders))) vprod = map(x -> vector + orders[end] * (1-scale) * x, vertices)
points = join([@sprintf("%.3f %.3f", real(v), imag(v)) for v in vprod], " ")
print(fh, "<polygon points=\"$points\" transform=\"translate($dim,$dim) rotate(-18)\" />\n")
end
print(fh, "</svg>") close(fh)</lang>
Kotlin
<lang scala>// version 1.1.2
import java.awt.* import java.awt.geom.Path2D import java.util.Random import javax.swing.*
class SierpinskiPentagon : JPanel() {
// exterior angle private val degrees072 = Math.toRadians(72.0)
/* After scaling we'll have 2 sides plus a gap occupying the length
of a side before scaling. The gap is the base of an isosceles triangle
with a base angle of 72 degrees. */
private val scaleFactor = 1.0 / (2.0 + Math.cos(degrees072) * 2.0)
private val margin = 20 private var limit = 0 private val r = Random()
init {
preferredSize = Dimension(640, 640)
background = Color.white
Timer(3000) {
limit++
if (limit >= 5) limit = 0
repaint()
}.start()
}
private fun drawPentagon(g: Graphics2D, x: Double, y: Double, s: Double, depth: Int) {
var angle = 3.0 * degrees072 // starting angle
var xx = x
var yy = y
var side = s
if (depth == 0) {
val p = Path2D.Double()
p.moveTo(xx, yy)
// draw from the top
for (i in 0 until 5) {
xx += Math.cos(angle) * side
yy -= Math.sin(angle) * side
p.lineTo(xx, yy)
angle += degrees072
}
g.color = RandomHue.next()
g.fill(p)
}
else {
side *= scaleFactor
/* Starting at the top of the highest pentagon, calculate
the top vertices of the other pentagons by taking the
length of the scaled side plus the length of the gap. */
val distance = side + side * Math.cos(degrees072) * 2.0
/* The top positions form a virtual pentagon of their own,
so simply move from one to the other by changing direction. */
for (i in 0 until 5) {
xx += Math.cos(angle) * distance
yy -= Math.sin(angle) * distance
drawPentagon(g, xx, yy, side, depth - 1)
angle += degrees072
}
}
}
override fun paintComponent(gg: Graphics) {
super.paintComponent(gg)
val g = gg as Graphics2D
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON)
val hw = width / 2
val radius = hw - 2.0 * margin
val side = radius * Math.sin(Math.PI / 5.0) * 2.0
drawPentagon(g, hw.toDouble(), 3.0 * margin, side, limit)
}
private class RandomHue {
/* Try to avoid random color values clumping together */
companion object {
val goldenRatioConjugate = (Math.sqrt(5.0) - 1.0) / 2.0
var hue = Math.random()
fun next(): Color {
hue = (hue + goldenRatioConjugate) % 1
return Color.getHSBColor(hue.toFloat(), 1.0f, 1.0f)
}
}
}
}
fun main(args: Array<String>) {
SwingUtilities.invokeLater {
val f = JFrame()
f.defaultCloseOperation = JFrame.EXIT_ON_CLOSE
f.title = "Sierpinski Pentagon"
f.isResizable = true
f.add(SierpinskiPentagon(), BorderLayout.CENTER)
f.pack()
f.setLocationRelativeTo(null)
f.isVisible = true
}
}</lang>
Mathematica
<lang mathematica>pentaFlake[0] = RegularPolygon[5]; pentaFlake[n_] :=
GeometricTransformation[pentaFlake[n - 1],
TranslationTransform /@
CirclePoints[{GoldenRatio^(2 n - 1), Pi/10}, 5]]
Graphics@pentaFlake[4]</lang>
- Output:
https://i.imgur.com/rvXvQc0.png
MATLAB
<lang MATLAB>[x, x0] = deal(exp(1i*(0.5:.4:2.1)*pi)); for k = 1 : 4
x = x(:) + x0 * (1 + sqrt(5)) * (3 + sqrt(5)) ^(k - 1) / 2 ^ k;
end patch('Faces', reshape(1 : 5 * 5 ^ k, 5, )', 'Vertices', [real(x(:)) imag(x(:))]) axis image off</lang>
- Output:
http://i.imgur.com/8ht6HqG.png
Perl
<lang perl>use ntheory qw(todigits); use Math::Complex;
$sides = 5; $order = 5; $dim = 250; $scale = ( 3 - 5**.5 ) / 2; push @orders, ((1 - $scale) * $dim) * $scale ** $_ for 0..$order-1;
open $fh, '>', 'sierpinski_pentagon.svg'; print $fh qq|<svg height="@{[$dim*2]}" width="@{[$dim*2]}" style="fill:blue" version="1.1" xmlns="http://www.w3.org/2000/svg">\n|;
$tau = 2 * 4*atan2(1, 1); push @vertices, cis( $_ * $tau / $sides ) for 0..$sides-1;
for $i (0 .. -1+$sides**$order) {
@base5 = todigits($i,5); @i = ( ((0)x(-1+$sides-$#base5) ), @base5); @v = @vertices[@i]; $vector = 0; $vector += $v[$_] * $orders[$_] for 0..$#orders;
my @points;
for (@vertices) {
$v = $vector + $orders[-1] * (1 - $scale) * $_;
push @points, sprintf '%.3f %.3f', $v->Re, $v->Im;
}
print $fh pgon(@points);
}
sub cis { Math::Complex->make(cos($_[0]), sin($_[0])) } sub pgon { my(@q)=@_; qq|<polygon points="@q" transform="translate($dim,$dim) rotate(-18)"/>\n| }
print $fh '</svg>'; close $fh;</lang> Sierpinski pentagon (offsite image)
Perl 6
<lang perl6>constant $sides = 5; constant order = 5; constant $dim = 250; constant scaling-factor = ( 3 - 5**.5 ) / 2; my @orders = ((1 - scaling-factor) * $dim) «*» scaling-factor «**» (^order);
my $fh = open('sierpinski_pentagon.svg', :w);
$fh.say: qq|<svg height="{$dim*2}" width="{$dim*2}" style="fill:blue" version="1.1" xmlns="http://www.w3.org/2000/svg">|;
my @vertices = map { cis( $_ * τ / $sides ) }, ^$sides;
for 0 ..^ $sides ** order -> $i {
my $vector = [+] @vertices[$i.base($sides).fmt("%{order}d").comb] «*» @orders;
$fh.say: pgon ((@orders[*-1] * (1 - scaling-factor)) «*» @vertices «+» $vector)».reals».fmt("%0.3f");
};
sub pgon (@q) { qq|<polygon points="{@q}" transform="translate({$dim},{$dim}) rotate(-18)"/>| }
$fh.say: '</svg>'; $fh.close;</lang>
Python
Draws the result on a canvas. Runs pretty slowly.
<lang python>from turtle import * import math speed(0) # 0 is the fastest speed. Otherwise, 1 (slow) to 10 (fast) hideturtle() # hide the default turtle
part_ratio = 2 * math.cos(math.radians(72)) side_ratio = 1 / (part_ratio + 2)
hide_turtles = True # show/hide turtles as they draw path_color = "black" # path color fill_color = "black" # fill color
- turtle, size
def pentagon(t, s):
t.color(path_color, fill_color) t.pendown() t.right(36) t.begin_fill() for i in range(5): t.forward(s) t.right(72) t.end_fill()
- iteration, turtle, size
def sierpinski(i, t, s):
t.setheading(0)
new_size = s * side_ratio
if i > 1:
i -= 1
# create four more turtles
for j in range(4):
t.right(36)
short = s * side_ratio / part_ratio
dist = [short, s, s, short][j]
# spawn a turtle
spawn = Turtle()
if hide_turtles:spawn.hideturtle()
spawn.penup()
spawn.setposition(t.position())
spawn.setheading(t.heading())
spawn.forward(dist)
# recurse for spawned turtles
sierpinski(i, spawn, new_size)
# recurse for parent turtle
sierpinski(i, t, new_size)
else:
# draw a pentagon
pentagon(t, s)
# delete turtle
del t
def main():
t = Turtle() t.hideturtle() t.penup() screen = t.getscreen() y = screen.window_height() t.goto(0, y/2-20) i = 5 # depth. i >= 1 size = 300 # side length # so the spawned turtles move only the distance to an inner pentagon size *= part_ratio # begin recursion sierpinski(i, t, size)
main()</lang>
See online implementation. See completed output.
Racket
<lang racket>#lang racket/base (require racket/draw pict racket/math racket/class)
- exterior angle
(define 72-degrees (degrees->radians 72))
- After scaling we'll have 2 sides plus a gap occupying the length
- of a side before scaling. The gap is the base of an isosceles triangle
- with a base angle of 72 degrees.
(define scale-factor (/ (+ 2 (* (cos 72-degrees) 2))))
- Starting at the top of the highest pentagon, calculate
- the top vertices of the other pentagons by taking the
- length of the scaled side plus the length of the gap.
(define dist-factor (+ 1 (* (cos 72-degrees) 2)))
- don't use scale, since it scales brushes too (making lines all tiny)
(define (draw-pentagon x y side depth dc)
(let recur ((x x) (y y) (side side) (depth depth))
(cond
[(zero? depth)
(define p (new dc-path%))
(send p move-to x y)
(for/fold ((x x) (y y) (α (* 3 72-degrees))) ((i 5))
(send p line-to x y)
(values (+ x (* side (cos α)))
(- y (* side (sin α)))
(+ α 72-degrees)))
(send p close)
(send dc draw-path p)]
[else
(define side/ (* side scale-factor))
(define dist (* side/ dist-factor))
;; The top positions form a virtual pentagon of their own,
;; so simply move from one to the other by changing direction.
(for/fold ((x x) (y y) (α (* 3 72-degrees))) ((i 5))
(recur x y side/ (sub1 depth))
(values (+ x (* dist (cos α)))
(- y (* dist (sin α)))
(+ α 72-degrees)))])))
(define (dc-draw-pentagon depth w h #:margin (margin 4))
(dc (lambda (dc dx dy)
(define old-brush (send dc get-brush))
(send dc set-brush (make-brush #:style 'transparent))
(draw-pentagon (/ w 2)
(* 3 margin)
(* (- (/ w 2) (* 2 margin))
(sin (/ pi 5)) 2)
depth
dc)
(send dc set-brush old-brush))
w h))
(dc-draw-pentagon 1 120 120) (dc-draw-pentagon 2 120 120) (dc-draw-pentagon 3 120 120) (dc-draw-pentagon 4 120 120) (dc-draw-pentagon 5 640 640)</lang>
Scala
Java Swing Interoperability
<lang Scala>import java.awt._ import java.awt.event.ActionEvent import java.awt.geom.Path2D
import javax.swing._
import scala.annotation.tailrec import scala.math.{Pi, cos, sin, sqrt}
object SierpinskiPentagon extends App {
SwingUtilities.invokeLater(() => {
class SierpinskiPentagon extends JPanel {
/* Try to avoid random color values clumping together */
private var hue = math.random
// exterior angle
private val deg072 = 2 * Pi / 5d //toRadians(72)
/* After scaling we'll have 2 sides plus a gap occupying the length
of a side before scaling. The gap is the base of an isosceles triangle
with a base angle of 72 degrees. */
//private val scaleFactor = 1 / (2 + cos(deg072) * 2)
private var limit = 0
private def drawPentagon(g: Graphics2D, x: Double, y: Double, side: Double, depth: Int): Unit = {
val scaleFactor = 1 / (2 + cos(deg072) * 2)
if (depth == 0) {
// draw from the top
@tailrec
def iter0(i: Int, x: Double, y: Double, angle: Double, p: Path2D.Double): Path2D.Double = {
if (i < 0) p
else {
p.lineTo(x, y)
iter0(i - 1, x + cos(angle) * side, y - sin(angle) * side, angle + deg072, p)
}
}
def p1: Path2D.Double = iter0(4, x, y, 3 * deg072, {
val p = new Path2D.Double
p.moveTo(x, y)
p
})
def p: Path2D.Double = iter0(4, x, y, 3 * deg072, p1)
def next: Color = {
hue = (hue + (sqrt(5) - 1) / 2) % 1
Color.getHSBColor(hue.toFloat, 1, 1)
}
g.setColor(next)
g.fill(p)
}
else {
val _side = side * scaleFactor
/* Starting at the top of the highest pentagon, calculate
the top vertices of the other pentagons by taking the
length of the scaled side plus the length of the gap. */
val distance = _side + _side * cos(deg072) * 2
/* The top positions form a virtual pentagon of their own,
so simply move from one to the other by changing direction. */
def iter1(i: Int, x: Double, y: Double, angle: Double): Unit = {
if (i < 0) ()
else {
drawPentagon(g, x, y, _side, depth - 1)
iter1(i - 1, x + cos(angle) * distance, y - sin(angle) * distance, angle + deg072)
}
}
iter1(4, x + cos(3 * deg072) * distance, y - sin(3 * deg072) * distance, 4 * deg072)
}
}
override def paintComponent(gg: Graphics): Unit = {
val (g, margin) = (gg.asInstanceOf[Graphics2D], 20)
val side = (getWidth / 2 - 2 * margin) * sin(Pi / 5) * 2
super.paintComponent(gg)
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON)
drawPentagon(g, getWidth / 2, 3 * margin, side, limit)
}
new Timer(3000, (_: ActionEvent) => {
limit += 1
if (limit >= 5) limit = 0
repaint()
}).start()
setPreferredSize(new Dimension(640, 640))
setBackground(Color.white)
}
val f = new JFrame("Sierpinski Pentagon") {
setDefaultCloseOperation(WindowConstants.EXIT_ON_CLOSE)
setResizable(true)
add(new SierpinskiPentagon, BorderLayout.CENTER)
pack()
setLocationRelativeTo(null)
setVisible(true)
}
})
}</lang>
Sidef
Generates a SVG image to STDOUT. Redirect to a file to capture and display it. <lang ruby>define order = 5 define sides = 5 define dim = 500 define scaling_factor = ((3 - 5**0.5) / 2) var orders = order.of {|i| ((1-scaling_factor) * dim) * scaling_factor**i }
say <<"STOP"; <?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 height="#{dim*2}" width="#{dim*2}"
style="fill:blue" transform="translate(#{dim},#{dim}) rotate(-18)"
version="1.1" xmlns="http://www.w3.org/2000/svg">
STOP
var vertices = sides.of {|i| Complex(0, i * Number.tau / sides).exp }
for i in ^(sides**order) {
var vector = ([vertices["%#{order}d" % i.base(sides) -> chars]] »*« orders «+»)
var points = (vertices »*» orders[-1]*(1-scaling_factor) »+» vector »reals()» «%« '%0.3f')
say ('<polygon points="' + points.join(' ') + '"/>')
} say '</svg>'</lang>
zkl
<lang zkl>const order=5, sides=5, dim=250, scaleFactor=((3.0 - (5.0).pow(0.5))/2); const tau=(0.0).pi*2; // 2*pi*r orders:=order.pump(List,fcn(n){ (1.0 - scaleFactor)*dim*scaleFactor.pow(n) });
println(
- <<<
0'|<?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 height="%d" width="%d" style="fill:blue" transform="translate(%d,%d) rotate(-18)"
version="1.1" xmlns="http://www.w3.org/2000/svg">|
- <<<
.fmt(dim*2,dim*2,dim,dim));
vertices:=sides.pump(List,fcn(s){ (1.0).toRectangular(tau*s/sides) }); // points on unit circle vx:=vertices.apply('wrap([(a,b)]v,x){ return(a*x,b*x) }, // scaled points orders[-1]*(1.0 - scaleFactor)); fmt:="%%0%d.%dB".fmt(sides,order).fmt; //-->%05.5B (leading zeros, 5 places, base 5) sides.pow(order).pump(Console.println,'wrap(i){
vector:=fmt(i).pump(List,vertices.get) // "00012"-->(vertices[0],..,vertices[2])
.zipWith(fcn([(a,b)]v,x){ return(a*x,b*x) },orders) // ((a,b)...)*x -->((ax,bx)...)
.reduce(fcn(vsum,v){ vsum[0]+=v[0]; vsum[1]+=v[1]; vsum },L(0.0, 0.0)); //-->(x,y)
pgon(vx.apply(fcn([(a,b)]v,c,d){ return(a+c,b+d) },vector.xplode()));
}); println("</svg>"); // 3,131 lines
fcn pgon(vertices){ // eg ( ((250,0),(248.595,1.93317),...), len 5
0'|<polygon points="%s"/>|.fmt(
vertices.pump(String,fcn(v){ "%.3f %.3f ".fmt(v.xplode()) }) )
}</lang>
- Output:
See this image. Displays fine in FireFox, in Chrome, it doesn't appear to be transformed so you only see part of the image.
zkl bbb > sierpinskiPentagon.zkl.svg $ wc sierpinskiPentagon.zkl.svg 3131 37519 314183 sierpinskiPentagon.zkl.svg