# 如何用Python画各种著名数学图案 | 附图+代码

Mandelbrot 集

'''

A fast Mandelbrot set wallpaper renderer

'''

import numpy as np

from PIL import Image

from numba import jit

MAXITERS = 200

@jit

def color(z, i):

v = np.log2(i + 1 - np.log2(np.log2(abs(z)))) / 5

if v < 1.0:

return v**4, v**2.5, v

else:

v = max(0, 2-v)

return v, v**1.5, v**3

@jit

def iterate(c):

z = 0j

for i in range(MAXITERS):

if z.real*z.real + z.imag*z.imag > RADIUS:

return color(z, i)

z = z*z + c

return 0, 0 ,0

def main(xmin, xmax, ymin, ymax, width, height):

x = np.linspace(xmin, xmax, width)

y = np.linspace(ymax, ymin, height)

z = x[None, :] + y[:, None]*1j

red, green, blue = np.asarray(np.frompyfunc(iterate, 1, 3)(z)).astype(np.float)

img = np.dstack((red, green, blue))

Image.fromarray(np.uint8(img*255)).save('mandelbrot.png')

if __name__ == '__main__':

main(-2.1, 0.8, -1.16, 1.16, 1200, 960)

'''

A kaleidoscope pattern with icosahedral symmetry.

'''

import numpy as np

from PIL import Image

from matplotlib.colors import hsv_to_rgb

def Klein(z):

'''Klein's j-function'''

return 1728 * (z * (z**10 + 11 * z**5 - 1))**5 / \

(-(z**20 + 1) + 228 * (z**15 - z**5) - 494 * z**10)**3

def RiemannSphere(z):

'''

map the complex plane to Riemann's sphere via stereographic projection

'''

t = 1 + z.real*z.real + z.imag*z.imag

return 2*z.real/t, 2*z.imag/t, 2/t-1

def Mobius(z):

'''

distort the result image by a mobius transformation

'''

return (z - 20)/(3*z + 1j)

def main(imgsize):

x = np.linspace(-6, 6, imgsize)

y = np.linspace(6, -6, imgsize)

z = x[None, :] + y[:, None]*1j

z = RiemannSphere(Klein(Mobius(Klein(z))))

# define colors in hsv space

H = np.sin(z[0]*np.pi)**2

S = np.cos(z[1]*np.pi)**2

V = abs(np.sin(z[2]*np.pi) * np.cos(z[2]*np.pi))**0.2

HSV = np.dstack((H, S, V))

# transform to rgb space

img = hsv_to_rgb(HSV)

Image.fromarray(np.uint8(img*255)).save('kaleidoscope.png')

if __name__ == '__main__':

import time

start = time.time()

main(imgsize=800)

end = time.time()

print('runtime: {:3f} seconds'.format(end - start))

Newton 迭代分形

import numpy as np

import matplotlib.pyplot as plt

from numba import jit

# define functions manually, do not use numpy's poly1d funciton!

@jit('complex64(complex64)', nopython=True)

def f(z):

# z*z*z is faster than z**3

return z*z*z - 1

@jit('complex64(complex64)', nopython=True)

def df(z):

return 3*z*z

@jit('float64(complex64)', nopython=True)

def iterate(z):

num = 0

while abs(f(z)) > 1e-4:

w = z - f(z)/df(z)

num += np.exp(-1/abs(w-z))

z = w

return num

def render(imgsize):

x = np.linspace(-1, 1, imgsize)

y = np.linspace(1, -1, imgsize)

z = x[None, :] + y[:, None] * 1j

img = np.frompyfunc(iterate, 1, 1)(z).astype(np.float)

fig = plt.figure(figsize=(imgsize/100.0, imgsize/100.0), dpi=100)

ax = fig.add_axes([0, 0, 1, 1], aspect=1)

ax.axis('off')

ax.imshow(img, cmap='hot')

fig.savefig('newton.png')

if __name__ == '__main__':

import time

start = time.time()

render(imgsize=400)

end = time.time()

print('runtime: {:03f} seconds'.format(end - start))

https://github.com/neozhaoliang/pywonderland/blob/master/src/misc/modulargroup.py

https://github.com/neozhaoliang/pywonderland/blob/master/src/misc/penrose.py

Wilson 算法

120 胞腔

# pylint: disable=unused-import

# pylint: disable=undefined-variable

from itertools import combinations, product

import numpy as np

from vapory import *

class Penrose(object):

GRIDS = [np.exp(2j * np.pi * i / 5) for i in range(5)]

def __init__(self, num_lines, shift, thin_color, fat_color, **config):

self.num_lines = num_lines

self.shift = shift

self.thin_color = thin_color

self.fat_color = fat_color

self.objs = self.compute_pov_objs(**config)

def compute_pov_objs(self, **config):

objects_pool = []

for rhombi, color in self.tile():

p1, p2, p3, p4 = rhombi

polygon = Polygon(5, p1, p2, p3, p4, p1,

Texture(Pigment('color', color), config['default']))

objects_pool.append(polygon)

for p, q in zip(rhombi, [p2, p3, p4, p1]):

cylinder = Cylinder(p, q, config['edge_thickness'], config['edge_texture'])

objects_pool.append(cylinder)

for point in rhombi:

x, y = point

sphere = Sphere((x, y, 0), config['vertex_size'], config['vertex_texture'])

objects_pool.append(sphere)

return Object(Union(*objects_pool))

def rhombus(self, r, s, kr, ks):

if (s - r)**2 % 5 == 1:

color = self.thin_color

else:

color = self.fat_color

point = (Penrose.GRIDS[r] * (ks - self.shift[s])

- Penrose.GRIDS[s] * (kr - self.shift[r])) *1j / Penrose.GRIDS[s-r].imag

index = [np.ceil((point/grid).real + shift)

for grid, shift in zip(Penrose.GRIDS, self.shift)]

vertices = []

for index[r], index[s] in [(kr, ks), (kr+1, ks), (kr+1, ks+1), (kr, ks+1)]:

vertices.append(np.dot(index, Penrose.GRIDS))

vertices_real = [(z.real, z.imag) for z in vertices]

return vertices_real, color

def tile(self):

for r, s in combinations(range(5), 2):

for kr, ks in product(range(-self.num_lines, self.num_lines+1), repeat=2):

yield self.rhombus(r, s, kr, ks)

def put_objs(self, *args):

return Object(self.objs, *args)

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