#!/usr/bin/env python
# coding: UTF-8
#
## @package _16_sierpinski
#
# Draws a Sierpinski Gasket.
#
# Usage: _16_sierpinski [number_of_divisions]
#
# @author Paulo Roma Cavalcanti
# @since 09/02/2016
# @see http://paulbourke.net/fractals/gasket/
# @see https://en.wikipedia.org/wiki/Sierpinski_triangle
# @see http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/siertri.htm
# @see http://www.oftenpaper.net/sierpinski.htm
import sys
try:
from tkinter import * # python 3
except ImportError:
from Tkinter import * # python 2
## Creates a Sierpinski Gasket, by recursively partitioning
# an initial triangle (a,b,c) into three or four new triangles.
#
# - There will be:
# - @f$3^{count}@f$ red triangles or
# - @f$4^{count}@f$ if the fourth triangle is drawn.
#
# - The number of white triangles is a Geometric Progression, starting at 1 and with ratio 3, given by:
# - P(0) = 0
# - P(n) = 3 * P(n-1) + 1
# - P(n) = @f$\frac{(3^n-1)}{2}.@f$
#
# @param a first vertex coordinates.
# @param b second vertex coordinates.
# @param c third vertex coordinates.
# @param n number of subdivisions on each edge (depth of recursion).
# @param fourth whether to add the fourth triangle.
#
def Sierpinski(a, b, c, n, fourth=False):
## list of points (a set of triangles).
points = []
## Pushes three vertices to the list of points.
def triangle( a, b, c ):
points.append ( a )
points.append ( b )
points.append ( c )
## Returns a point on segment p1-p2 corresponding to parameter s.
def mix(p1,p2,s):
return [(p1[0]+p2[0])*s, (p1[1]+p2[1])*s]
## Divides triangle (a,b,c), by recursively subdividing
# each edge at its middle point, and connecting the new points.
def divideTriangle( a, b, c, count ):
# check for end of recursion
if ( count == 0 ):
triangle( a, b, c )
else:
# bisect the sides
ab = mix( a, b, 0.5 )
ac = mix( a, c, 0.5 )
bc = mix( b, c, 0.5 )
count -= 1
# three new triangles
divideTriangle( a, ab, ac, count )
divideTriangle( c, ac, bc, count )
divideTriangle( b, bc, ab, count )
# add the middle triangle
if ( fourth ): divideTriangle( ac, bc, ab, count )
divideTriangle (a, b, c, n)
return points
## Draw triangles, given in points, on canvas c.
def draw(c, points, contour=False):
w = c.winfo_width()//2
h = c.winfo_height()//2
centerx = w
centery = h
c.delete(ALL)
for i in range(0,len(points),3):
if ( not contour ):
c.create_polygon(centerx+w*points[i] [0],centery-h*points[i ][1],
centerx+w*points[i+1][0],centery-h*points[i+1][1],
centerx+w*points[i+2][0],centery-h*points[i+2][1], fill='red', outline='black')
else:
c.create_line(centerx+w*points[i] [0],centery-h*points[i ][1],
centerx+w*points[i+1][0],centery-h*points[i+1][1],
centerx+w*points[i+2][0],centery-h*points[i+2][1],
centerx+w*points[i] [0],centery-h*points[i ][1], fill='black', width=2)
## Main program.
def main(argv=None):
## Resize the graphics when the window changes.
def resize(event=None):
global pts
draw(canvas, pts, cntVar.get()=='ON')
## Redraw the graphics when the scale changes
def redraw(event=None):
global pts
# Initialize the corners of the gasket with three points.
vertices = [
[ -1, -1 ],
[ 0, 1 ],
[ 1, -1 ]
]
pg = lambda n: (3**n-1)//2
ndiv = slider.get()
pts = Sierpinski( vertices[0], vertices[1], vertices[2], ndiv, fillVar.get()=='ON' )
lr.configure(text="Red Triangles = %d "%(len(pts)//3))
la.configure(text=" White Triangles = %d"%(pg(ndiv)))
resize(event)
ndiv = 2
if argv is None:
argv = sys.argv
if len(argv) > 1:
try:
ndiv = abs(int(argv[1]))
except:
ndiv = 3
root = Tk()
root.title("Sierpinski Gasket")
bot = Frame(root)
bot.pack(side=BOTTOM)
top = Frame(root)
top.pack(side=TOP)
canvas = Canvas(root,width=400,height=400);
canvas.bind("<Configure>", resize)
canvas.pack(expand=YES,fill=BOTH)
slider = Scale(bot, from_=0, to=8, command=redraw, orient=HORIZONTAL)
slider.set(ndiv)
slider.pack(side=LEFT, anchor = W)
cntVar = StringVar() # create a checkbutton for the drawing style
cntVar.set ( "OFF" )
c = Checkbutton (bot, text="Contour", variable=cntVar, onvalue="ON", offvalue="OFF", command=resize)
c.pack(side=BOTTOM)
fillVar = StringVar() # create another checkbutton for the number of triangles
fillVar.set ( "OFF" )
c = Checkbutton (bot, text="Fill", variable=fillVar, onvalue="ON", offvalue="OFF", command=redraw)
c.pack(side=LEFT)
lr = Label(top, text="")
lr.pack(side=LEFT,anchor = W)
la = Label(top, text="")
la.pack(side=RIGHT,anchor = E)
redraw(None)
mainloop()
if __name__=='__main__':
sys.exit(main())
