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New fractals folder (#4277)

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* Move koch_snowflake.py to fractals-folder

* Move mandelbrot.py to fractals-folder

* Move sierpinski_triangle.py to fractals-folder
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fractals/koch_snowflake.py Normal file
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"""
Description
The Koch snowflake is a fractal curve and one of the earliest fractals to
have been described. The Koch snowflake can be built up iteratively, in a
sequence of stages. The first stage is an equilateral triangle, and each
successive stage is formed by adding outward bends to each side of the
previous stage, making smaller equilateral triangles.
This can be achieved through the following steps for each line:
1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1
as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.
(description adapted from https://en.wikipedia.org/wiki/Koch_snowflake )
(for a more detailed explanation and an implementation in the
Processing language, see https://natureofcode.com/book/chapter-8-fractals/
#84-the-koch-curve-and-the-arraylist-technique )
Requirements (pip):
- matplotlib
- numpy
"""
from __future__ import annotations
import matplotlib.pyplot as plt # type: ignore
import numpy
# initial triangle of Koch snowflake
VECTOR_1 = numpy.array([0, 0])
VECTOR_2 = numpy.array([0.5, 0.8660254])
VECTOR_3 = numpy.array([1, 0])
INITIAL_VECTORS = [VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]
# uncomment for simple Koch curve instead of Koch snowflake
# INITIAL_VECTORS = [VECTOR_1, VECTOR_3]
def iterate(initial_vectors: list[numpy.ndarray], steps: int) -> list[numpy.ndarray]:
"""
Go through the number of iterations determined by the argument "steps".
Be careful with high values (above 5) since the time to calculate increases
exponentially.
>>> iterate([numpy.array([0, 0]), numpy.array([1, 0])], 1)
[array([0, 0]), array([0.33333333, 0. ]), array([0.5 , \
0.28867513]), array([0.66666667, 0. ]), array([1, 0])]
"""
vectors = initial_vectors
for i in range(steps):
vectors = iteration_step(vectors)
return vectors
def iteration_step(vectors: list[numpy.ndarray]) -> list[numpy.ndarray]:
"""
Loops through each pair of adjacent vectors. Each line between two adjacent
vectors is divided into 4 segments by adding 3 additional vectors in-between
the original two vectors. The vector in the middle is constructed through a
60 degree rotation so it is bent outwards.
>>> iteration_step([numpy.array([0, 0]), numpy.array([1, 0])])
[array([0, 0]), array([0.33333333, 0. ]), array([0.5 , \
0.28867513]), array([0.66666667, 0. ]), array([1, 0])]
"""
new_vectors = []
for i, start_vector in enumerate(vectors[:-1]):
end_vector = vectors[i + 1]
new_vectors.append(start_vector)
difference_vector = end_vector - start_vector
new_vectors.append(start_vector + difference_vector / 3)
new_vectors.append(
start_vector + difference_vector / 3 + rotate(difference_vector / 3, 60)
)
new_vectors.append(start_vector + difference_vector * 2 / 3)
new_vectors.append(vectors[-1])
return new_vectors
def rotate(vector: numpy.ndarray, angle_in_degrees: float) -> numpy.ndarray:
"""
Standard rotation of a 2D vector with a rotation matrix
(see https://en.wikipedia.org/wiki/Rotation_matrix )
>>> rotate(numpy.array([1, 0]), 60)
array([0.5 , 0.8660254])
>>> rotate(numpy.array([1, 0]), 90)
array([6.123234e-17, 1.000000e+00])
"""
theta = numpy.radians(angle_in_degrees)
c, s = numpy.cos(theta), numpy.sin(theta)
rotation_matrix = numpy.array(((c, -s), (s, c)))
return numpy.dot(rotation_matrix, vector)
def plot(vectors: list[numpy.ndarray]) -> None:
"""
Utility function to plot the vectors using matplotlib.pyplot
No doctest was implemented since this function does not have a return value
"""
# avoid stretched display of graph
axes = plt.gca()
axes.set_aspect("equal")
# matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
# y-coordinates as inputs, which are constructed from the vector-list using
# zip()
x_coordinates, y_coordinates = zip(*vectors)
plt.plot(x_coordinates, y_coordinates)
plt.show()
if __name__ == "__main__":
import doctest
doctest.testmod()
processed_vectors = iterate(INITIAL_VECTORS, 5)
plot(processed_vectors)

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"""
The Mandelbrot set is the set of complex numbers "c" for which the series
"z_(n+1) = z_n * z_n + c" does not diverge, i.e. remains bounded. Thus, a
complex number "c" is a member of the Mandelbrot set if, when starting with
"z_0 = 0" and applying the iteration repeatedly, the absolute value of
"z_n" remains bounded for all "n > 0". Complex numbers can be written as
"a + b*i": "a" is the real component, usually drawn on the x-axis, and "b*i"
is the imaginary component, usually drawn on the y-axis. Most visualizations
of the Mandelbrot set use a color-coding to indicate after how many steps in
the series the numbers outside the set diverge. Images of the Mandelbrot set
exhibit an elaborate and infinitely complicated boundary that reveals
progressively ever-finer recursive detail at increasing magnifications, making
the boundary of the Mandelbrot set a fractal curve.
(description adapted from https://en.wikipedia.org/wiki/Mandelbrot_set )
(see also https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set )
"""
import colorsys
from PIL import Image # type: ignore
def get_distance(x: float, y: float, max_step: int) -> float:
"""
Return the relative distance (= step/max_step) after which the complex number
constituted by this x-y-pair diverges. Members of the Mandelbrot set do not
diverge so their distance is 1.
>>> get_distance(0, 0, 50)
1.0
>>> get_distance(0.5, 0.5, 50)
0.061224489795918366
>>> get_distance(2, 0, 50)
0.0
"""
a = x
b = y
for step in range(max_step):
a_new = a * a - b * b + x
b = 2 * a * b + y
a = a_new
# divergence happens for all complex number with an absolute value
# greater than 4
if a * a + b * b > 4:
break
return step / (max_step - 1)
def get_black_and_white_rgb(distance: float) -> tuple:
"""
Black&white color-coding that ignores the relative distance. The Mandelbrot
set is black, everything else is white.
>>> get_black_and_white_rgb(0)
(255, 255, 255)
>>> get_black_and_white_rgb(0.5)
(255, 255, 255)
>>> get_black_and_white_rgb(1)
(0, 0, 0)
"""
if distance == 1:
return (0, 0, 0)
else:
return (255, 255, 255)
def get_color_coded_rgb(distance: float) -> tuple:
"""
Color-coding taking the relative distance into account. The Mandelbrot set
is black.
>>> get_color_coded_rgb(0)
(255, 0, 0)
>>> get_color_coded_rgb(0.5)
(0, 255, 255)
>>> get_color_coded_rgb(1)
(0, 0, 0)
"""
if distance == 1:
return (0, 0, 0)
else:
return tuple(round(i * 255) for i in colorsys.hsv_to_rgb(distance, 1, 1))
def get_image(
image_width: int = 800,
image_height: int = 600,
figure_center_x: float = -0.6,
figure_center_y: float = 0,
figure_width: float = 3.2,
max_step: int = 50,
use_distance_color_coding: bool = True,
) -> Image.Image:
"""
Function to generate the image of the Mandelbrot set. Two types of coordinates
are used: image-coordinates that refer to the pixels and figure-coordinates
that refer to the complex numbers inside and outside the Mandelbrot set. The
figure-coordinates in the arguments of this function determine which section
of the Mandelbrot set is viewed. The main area of the Mandelbrot set is
roughly between "-1.5 < x < 0.5" and "-1 < y < 1" in the figure-coordinates.
>>> get_image().load()[0,0]
(255, 0, 0)
>>> get_image(use_distance_color_coding = False).load()[0,0]
(255, 255, 255)
"""
img = Image.new("RGB", (image_width, image_height))
pixels = img.load()
# loop through the image-coordinates
for image_x in range(image_width):
for image_y in range(image_height):
# determine the figure-coordinates based on the image-coordinates
figure_height = figure_width / image_width * image_height
figure_x = figure_center_x + (image_x / image_width - 0.5) * figure_width
figure_y = figure_center_y + (image_y / image_height - 0.5) * figure_height
distance = get_distance(figure_x, figure_y, max_step)
# color the corresponding pixel based on the selected coloring-function
if use_distance_color_coding:
pixels[image_x, image_y] = get_color_coded_rgb(distance)
else:
pixels[image_x, image_y] = get_black_and_white_rgb(distance)
return img
if __name__ == "__main__":
import doctest
doctest.testmod()
# colored version, full figure
img = get_image()
# uncomment for colored version, different section, zoomed in
# img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
# figure_width = 0.8)
# uncomment for black and white version, full figure
# img = get_image(use_distance_color_coding = False)
# uncomment to save the image
# img.save("mandelbrot.png")
img.show()

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#!/usr/bin/python
"""Author Anurag Kumar | anuragkumarak95@gmail.com | git/anuragkumarak95
Simple example of Fractal generation using recursive function.
What is Sierpinski Triangle?
>>The Sierpinski triangle (also with the original orthography Sierpinski), also called
the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set
with the overall shape of an equilateral triangle, subdivided recursively into smaller
equilateral triangles. Originally constructed as a curve, this is one of the basic
examples of self-similar sets, i.e., it is a mathematically generated pattern that can
be reproducible at any magnification or reduction. It is named after the Polish
mathematician Wacław Sierpinski, but appeared as a decorative pattern many centuries
prior to the work of Sierpinski.
Requirements(pip):
- turtle
Python:
- 2.6
Usage:
- $python sierpinski_triangle.py <int:depth_for_fractal>
Credits: This code was written by editing the code from
http://www.riannetrujillo.com/blog/python-fractal/
"""
import sys
import turtle
PROGNAME = "Sierpinski Triangle"
points = [[-175, -125], [0, 175], [175, -125]] # size of triangle
def getMid(p1, p2):
return ((p1[0] + p2[0]) / 2, (p1[1] + p2[1]) / 2) # find midpoint
def triangle(points, depth):
myPen.up()
myPen.goto(points[0][0], points[0][1])
myPen.down()
myPen.goto(points[1][0], points[1][1])
myPen.goto(points[2][0], points[2][1])
myPen.goto(points[0][0], points[0][1])
if depth > 0:
triangle(
[points[0], getMid(points[0], points[1]), getMid(points[0], points[2])],
depth - 1,
)
triangle(
[points[1], getMid(points[0], points[1]), getMid(points[1], points[2])],
depth - 1,
)
triangle(
[points[2], getMid(points[2], points[1]), getMid(points[0], points[2])],
depth - 1,
)
if __name__ == "__main__":
if len(sys.argv) != 2:
raise ValueError(
"right format for using this script: "
"$python fractals.py <int:depth_for_fractal>"
)
myPen = turtle.Turtle()
myPen.ht()
myPen.speed(5)
myPen.pencolor("red")
triangle(points, int(sys.argv[1]))