Penrose Tilings

Penrose Tilings

Thesis

Penrose tilings cover the plane aperiodically with finitely many prototiles. Substitution counts track the golden ratio \(\phi\).

Mental model

  thick/thin substitution → ratio → φ

Worked examples

step thick thin total ratio_thick/thin
   0        1        0        1  inf
   1        2        1        3  2.0000
   2        5        3        8  1.6667
   3       13        8       21  1.6250
   4       34       21       55  1.6190
   5       89       55      144  1.6182
   6      233      144      377  1.6181
   7      610      377      987  1.6180
   8     1597      987     2584  1.6180
phi = 1.618033988749895 phi^2 = 2.618033988749895

Trap

Calling any non-repeating pattern Penrose. Matching rules matter.

Boring rule

Cite the substitution rule and \(\phi\), not vibes.

Try this

1. Draw three inflation steps.
2. Read about quasicrystals.
3. Compare Ammann–Beenker tilings.

Sources

• Penrose; Grünbaum & Shephard