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# Penrose tilings with Sage

I would like to make a poster from a Penrose tiling. Google didn't give me any Sage code for doing this, but I wonder if someone here knows of such a thing!

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## Comments

You could try http://preshing.com/20110831/penrose-tiling-explained. An internet search for "penrose tiling algorithm" seems likely to be fruitful.

Well, yes. I was hoping for some enterprising person to turn the explanation into code for me! ;) And in fact I'd like code which lets me color the tiling in some enlightening way.

Inputs would be size of the output image, and maybe some options for how the picture is colored (highlight symmetry, or highlight method of construction, perhaps).

## 1 Answer

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# Adapted from http://preshing.com/20110831/penrose-tiling-explained

golden_ratio = (1 + sqrt(5))/2

def subdivide(triangles):
result = []
for color, A, B, C in triangles:
if color == 0:
# Subdivide red triangle
P = A + (B - A) / golden_ratio
result += [(0, C, P, B), (1, P, C, A)]
else:
# Subdivide blue triangle
Q = B + (A - B) / golden_ratio
R = B + (C - B) / golden_ratio
result += [(1, R, C, A), (1, Q, R, B), (0, R, Q, A)]
return result

# Create wheel of red triangles around the origin
triangles = []
for i in xrange(10):
B = exp((2*i - 1)*pi/10*I)
C = exp((2*i + 1)*pi/10*I)
if i % 2 == 0:
B, C = C, B  # Make sure to mirror every second triangle
triangles.append((0, 0, B, C))

# Draw n iterates
n = 5
for i in range(n):
P = Graphics()
for verts in triangles:
#P += polygon([(z.real(), z.imag()) for z in verts[1:]], color=['red', 'blue'][verts])
P += line([(z.real(), z.imag()) for z in (verts, verts, verts)], color='black', thickness=2)
P.show(axes=False, aspect_ratio=1)
triangles = subdivide(triangles)

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## Comments

Nice code ! Actually, the code is much faster if you replace the symbolic constants (sqrt(5) and the exponential) by real numbers.

Thanks! I updated the code to allow various coloring schemes, draw circular sectors on the tiles, and output the graphics object so that differing levels of iteration can be overlaid: http://www.nilesjohnson.net/aperiodic-tilings.html

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Asked: 2012-10-22 16:42:41 +0200

Seen: 950 times

Last updated: Mar 23 '13