examples/menger.py

   1 #!/usr/bin/env python3
   2
   3 # Plot a Menger sponge.
   4 #
   5 # The construction of a Menger sponge can be described as follows:
   6 #
   7 # 1. Begin with a cube.
   8 # 2. Divide every face of the cube into 9 squares, like a Rubik's Cube. This
   9 #    will sub-divide the cube into 27 smaller cubes.
  10 # 3. Remove the smaller cube in the middle of each face, and remove the smaller
  11 #    cube in the very center of the larger cube, leaving 20 smaller cubes. This
  12 #    is a level-1 Menger sponge (resembling a Void Cube).
  13 # 4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue
  14 #    to iterate.
  15
  16 import argparse
  17
  18 import matplotlib.pyplot as plt
  19
  20 from math import ceil
  21
  22 from matplotlib import pylab
  23 from mpl_toolkits.mplot3d import Axes3D
  24
  25 from linpy import Le, Polyhedron, symbols
  26
  27
  28 x, y, z = symbols('x y z')
  29
  30 _x, _y, _z = x.asdummy(), y.asdummy(), z.asdummy()
  31
  32
  33 def translate(domain, *, dx=0, dy=0, dz=0):
  34     domain &= Polyhedron([x - _x + dx, y - _y + dy, z - _z + dz])
  35     domain = domain.project([x, y, z])
  36     domain = domain.subs({_x: x, _y: y, _z: z})
  37     return domain
  38
  39
  40 def _menger(domain, size):
  41     result = domain
  42     result |= translate(domain, dx=0, dy=size, dz=0)
  43     result |= translate(domain, dx=0, dy=2*size, dz=0)
  44     result |= translate(domain, dx=size, dy=0, dz=0)
  45     result |= translate(domain, dx=size, dy=2*size, dz=0)
  46     result |= translate(domain, dx=2*size, dy=0, dz=0)
  47     result |= translate(domain, dx=2*size, dy=size, dz=0)
  48     result |= translate(domain, dx=2*size, dy=2*size, dz=0)
  49     result |= translate(domain, dx=0, dy=0, dz=size)
  50     result |= translate(domain, dx=0, dy=2*size, dz=size)
  51     result |= translate(domain, dx=2*size, dy=0, dz=size)
  52     result |= translate(domain, dx=2*size, dy=2*size, dz=size)
  53     result |= translate(domain, dx=0, dy=0, dz=2*size)
  54     result |= translate(domain, dx=0, dy=size, dz=2*size)
  55     result |= translate(domain, dx=0, dy=2*size, dz=2*size)
  56     result |= translate(domain, dx=size, dy=0, dz=2*size)
  57     result |= translate(domain, dx=size, dy=2*size, dz=2*size)
  58     result |= translate(domain, dx=2*size, dy=0, dz=2*size)
  59     result |= translate(domain, dx=2*size, dy=size, dz=2*size)
  60     result |= translate(domain, dx=2*size, dy=2*size, dz=2*size)
  61     return result
  62
  63
  64 def menger(domain, count=1, cut=False):
  65     size = 1
  66     for i in range(count):
  67         domain = _menger(domain, size)
  68         size *= 3
  69     if cut:
  70         domain &= Le(x + y + z, ceil(3 * size / 2))
  71     return domain
  72
  73 if __name__ == '__main__':
  74     parser = argparse.ArgumentParser(
  75         description='Compute a Menger sponge.')
  76     parser.add_argument(
  77         '-n', '--iterations', type=int, default=2,
  78         help='number of iterations (default: 2)')
  79     parser.add_argument(
  80         '-c', '--cut', action='store_true', default=False,
  81         help='cut the sponge')
  82     args = parser.parse_args()
  83     cube = Le(0, x) & Le(x, 1) & Le(0, y) & Le(y, 1) & Le(0, z) & Le(z, 1)
  84     fractal = menger(cube, args.iterations, args.cut)
  85     fig = plt.figure(facecolor='white')
  86     plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
  87     fractal.plot(plot)
  88     pylab.show()