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Add Mandelbrot algorithm
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graphics/mandelbrot.py
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150
graphics/mandelbrot.py
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"""
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The Mandelbrot set is the set of complex numbers "c" for which the series
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"z_(n+1) = z_n * z_n + c" does not diverge, i.e. remains bounded. Thus, a
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complex number "c" is a member of the Mandelbrot set if, when starting with
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"z_0 = 0" and applying the iteration repeatedly, the absolute value of
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"z_n" remains bounded for all "n > 0". Complex numbers can be written as
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"a + b*i": "a" is the real component, usually drawn on the x-axis, and "b*i"
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is the imaginary component, usually drawn on the y-axis. Most visualizations
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of the Mandelbrot set use a color-coding to indicate after how many steps in
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the series the numbers outside the set diverge. Images of the Mandelbrot set
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exhibit an elaborate and infinitely complicated boundary that reveals
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progressively ever-finer recursive detail at increasing magnifications, making
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the boundary of the Mandelbrot set a fractal curve.
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(description adapted from https://en.wikipedia.org/wiki/Mandelbrot_set )
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(see also https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set )
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"""
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import colorsys
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from PIL import Image # type: ignore
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def getDistance(x: float, y: float, max_step: int) -> float:
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"""
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Return the relative distance (= step/max_step) after which the complex number
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constituted by this x-y-pair diverges. Members of the Mandelbrot set do not
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diverge so their distance is 1.
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>>> getDistance(0, 0, 50)
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1.0
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>>> getDistance(0.5, 0.5, 50)
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0.061224489795918366
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>>> getDistance(2, 0, 50)
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0.0
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"""
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a = x
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b = y
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for step in range(max_step):
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a_new = a * a - b * b + x
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b = 2 * a * b + y
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a = a_new
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# divergence happens for all complex number with an absolute value
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# greater than 4
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if a * a + b * b > 4:
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break
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return step / (max_step - 1)
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def get_black_and_white_rgb(distance: float) -> tuple:
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"""
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Black&white color-coding that ignores the relative distance. The Mandelbrot
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set is black, everything else is white.
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>>> get_black_and_white_rgb(0)
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(255, 255, 255)
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>>> get_black_and_white_rgb(0.5)
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(255, 255, 255)
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>>> get_black_and_white_rgb(1)
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(0, 0, 0)
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"""
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if distance == 1:
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return (0, 0, 0)
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else:
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return (255, 255, 255)
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def get_color_coded_rgb(distance: float) -> tuple:
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"""
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Color-coding taking the relative distance into account. The Mandelbrot set
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is black.
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>>> get_color_coded_rgb(0)
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(255, 0, 0)
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>>> get_color_coded_rgb(0.5)
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(0, 255, 255)
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>>> get_color_coded_rgb(1)
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(0, 0, 0)
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"""
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if distance == 1:
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return (0, 0, 0)
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else:
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return tuple(round(i * 255) for i in colorsys.hsv_to_rgb(distance, 1, 1))
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def get_image(
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image_width: int = 800,
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image_height: int = 600,
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figure_center_x: float = -0.6,
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figure_center_y: float = 0,
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figure_width: float = 3.2,
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max_step: int = 50,
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use_distance_color_coding: bool = True,
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) -> Image.Image:
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"""
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Function to generate the image of the Mandelbrot set. Two types of coordinates
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are used: image-coordinates that refer to the pixels and figure-coordinates
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that refer to the complex numbers inside and outside the Mandelbrot set. The
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figure-coordinates in the arguments of this function determine which section
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of the Mandelbrot set is viewed. The main area of the Mandelbrot set is
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roughly between "-1.5 < x < 0.5" and "-1 < y < 1" in the figure-coordinates.
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>>> get_image().load()[0,0]
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(255, 0, 0)
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>>> get_image(use_distance_color_coding = False).load()[0,0]
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(255, 255, 255)
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"""
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img = Image.new("RGB", (image_width, image_height))
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pixels = img.load()
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# loop through the image-coordinates
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for image_x in range(image_width):
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for image_y in range(image_height):
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# determine the figure-coordinates based on the image-coordinates
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figure_height = figure_width / image_width * image_height
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figure_x = figure_center_x + (image_x / image_width - 0.5) * figure_width
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figure_y = figure_center_y + (image_y / image_height - 0.5) * figure_height
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distance = getDistance(figure_x, figure_y, max_step)
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# color the corresponding pixel based on the selected coloring-function
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if use_distance_color_coding:
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pixels[image_x, image_y] = get_color_coded_rgb(distance)
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else:
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pixels[image_x, image_y] = get_black_and_white_rgb(distance)
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return img
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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# colored version, full figure
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img = get_image()
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# uncomment for colored version, different section, zoomed in
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# img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
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# figure_width = 0.8)
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# uncomment for black and white version, full figure
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# img = get_image(use_distance_color_coding = False)
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# uncomment to save the image
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# img.save("mandelbrot.png")
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img.show()
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