Chaos and Fractals
Sierpinski Triangle with a Chaos Game using Interactive Matplotlib
I am new to Fractals but i really love the idea of nature doing things we do not see as structured objects, but that indeed are.
Building figures like the following one often implies using feedback systems. You can think of a function that is executed on its own output for many iterations. The most common known figure is the Mandelbrot set.
Here we find a Fractal with the "Hipster-Object" the Triangle
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i had issues finding the right preparation for an interactive plot updating on each iteration: i found a solution here for ipython: http://stackoverflow.com/questions/21360361/how-to-dynamically-update-a-plot-in-a-loop-in-ipython-notebook-within-one-cell
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# the marked #! lines are essential for the interactivity
%matplotlib inline #!
import random
import numpy as np
import time
import pylab as pl #!
from IPython import display #!
import numpy as np
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pylab.rcParams['figure.figsize'] = (10.0, 8.0)
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def Sierpinski_Chaos(gp=None,n=None):
""" This Chaos game shows how randomness can lead to structure:
1. 3 points have to be drawn in the plane
2. Mark a triangle with these points with pl.plot
3. draw a random point outside this triangle- this is the first 'Game-Point(gp)'
Repeat til n:
4. Choose a randomly a base-point out of the three corner points
5. Build the vector between the 'Game-Point' and the randomly chosen base-point
and mark a point (scatter) half way to the base-point
Created 08-29-2014 by Pierre Noire """
#1.,2.
pl.plot([0,4,2,0],[0,0,4,0])
pl.xlim(0,4)
pl.ylim(0,4)
base_points=[[0,0],[4,0],[2,4]]
if gp==None:
gp=np.array([5,5])#starting game_point
if n==None:
n=500 #number of iterations
for n in range(500):
gp_log=gp.copy()
pl.scatter(gp[0],gp[1],lw='0',s=20)#3.
pl.xlim(0,4)#!
pl.ylim(0,4)#!
pl.draw()#!
display.clear_output(wait=True)#!
display.display(pl.gcf())#!
time.sleep(0.0000005)#!
#4
fort_wheel=random.choice(base_points)
rand_base=np.array(fort_wheel)
#5
gp=gp-1.0/2*(gp-rand_base)
gp_log=np.concatenate((gp_log,gp))
#(gp-rand_base) is
#"direction-vector" starting from the gp and just walking half way leads to new gp
return gp_log
To Read more about this Chaos Game look here:
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build_hipster=Sierpinski_Chaos(n=1500)
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from IPython.display import IFrame
IFrame('https://en.wikipedia.org/wiki/Chaos_game', width=700, height=500)
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#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py #install if you haven´t yet
%load_ext version_information
%reload_ext version_information
%version_information numpy, matplotlib,
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