Creating Fractals

Exercise

The purpose of the present exercise is to generate a fractal with the method of Iterated Maps. This method is demonstrated in the Game of Chaos for an example.

$P_0$ $x_0$ $y_0$ $x_k$ $y_k$ $P_k$ $x_{k-1}$ $y_{k-1}$ $P_{k-1}$ using the following formulas:

x_k = a * x_{k-1} + b * y_{k-1} + e

y_k = c * x_{k-1} + d * y_{k-1} + f

$a, b, c, d, e,$ $f$ are coefficient that are assigned in the table below.

	$a$	$b$	$c$	$d$	$e$	$f$
A	.85	.04	-.04	.85	.00	1.60
B	.20	-.26	.23	.22	.00	1.60
C	-.15	.28	.26	.24	.00	.44
D	.00	.00	.00	.16	.00	.00

Every time we generate a new point we select one of the rows at random in the table above and use the coefficients in that row to compute the coordinates of the point. The rows are not be selected with uniform probability, but with the following ones:

Pr(A) = .85
Pr(B) = .07
Pr(C) = .07
Pr(D) = .01

Finally, we then draw the last 10,000 point you generated.

Below is the code to execute the above detail exercise.

Generate Functions Which Do the Heavy Lifting


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from numpy.random import choice
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def select_locator():
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    """Method to select the row randomly (w/ weights)"""
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    locator_list = ["A", "B", "C", "D"]
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    locator = choice(locator_list, 1, p=[0.85, 0.07, 0.07, 0.01])
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    return locator[0]
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def get_coefficients(locator):
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    """Method to return the coefficients to use"""
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    if locator == "A":
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        a = .85
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        b = .04
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        c = -.04
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        d = .85
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        e = 0.0
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        f = 1.6
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        return a,b,c,d,e,f
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    elif locator == "B":
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        a = .2
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        b = -.26
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        c = .23
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        d = .22
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        e = 0.0
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        f = 1.6
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        return a,b,c,d,e,f
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    elif locator == "C":
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        a = -.15
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        b = .28
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        c = .26
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        d = .24
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        e = 0.0
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        f = .44
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        return a,b,c,d,e,f
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    elif locator == "D":
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        a = 0.0
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        b = 0.0
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        c = 0.0
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        d = .16
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        e = 0.0
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        f = 0.0
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        return a,b,c,d,e,f
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def get_new_points(a,b,c,d,current_x,current_y):
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    """A function to calculate the new points location 
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    based on the previous points location."""
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    new_x = a * current_x + b * current_y + e
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    new_y = c * current_x + d * current_y + f
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    return new_x, new_y

Checking the `select_locator()` function works properly via quick simulation...


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import pandas as pd
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import matplotlib.pyplot as plt
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import seaborn as sns
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# Simulate 10000 draws and plot the histgram
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options = []
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for x in range(10000):
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    options.append(select_locator())
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plt.figure(figsize=(6,6))
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_ = pd.Series(options).value_counts().plot(kind = 'bar', zorder=3)
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plt.grid()
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sns.despine()

Calculate New Coordinate Points


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# Create original points
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x_vals = [0]
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y_vals = [0]
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# Number of new points to run...
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new_points = 10100
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# Calculate all the new points...
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for x in range(new_points):
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    # Get coefficient locator
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    locator = select_locator()
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    # Get the coefficients
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    a,b,c,d,e,f = get_coefficients(locator)
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    # Get most recent coordinates
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    current_x = x_vals[-1]
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    current_y = y_vals[-1]
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    # Pass coefficients and coordinates to calculate the new locations
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    new_x, new_y = get_new_points(a,b,c,d,current_x,current_y)
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    x_vals.append(new_x)
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    y_vals.append(new_y)

Create Fractal Figure