Penrose Tiling: Aperiodic Order & Quasicrystals

Order without repetition

An ordinary tiling, like the square grid on a chessboard, is periodic: slide the whole pattern by a fixed vector and it lands perfectly back on itself. A Penrose tiling refuses to do this — no matter how far you shift it, it never coincides with the original. Yet it is not random either: it obeys rigid matching rules, usually shown as coloured arcs, that dictate how the two tile shapes may meet edge to edge. This combination of strict local rules with global non-repetition is called aperiodic order, and it overturned a long-held assumption that ordered matter must always be periodic.

The golden ratio and the P3 rhombi

The most common variant, P3, uses two rhombi: a thick rhombus with angles 72° and 108°, and a thin rhombus with angles 36° and 144° — both multiples of 36° = π/5, the fundamental angle of a regular pentagon. As a Penrose tiling grows, the ratio of thick to thin tiles converges to the golden ratio:

φ = (1 + √5) / 2 ≈ 1.618

Deflation rule (each step scales new tiles by 1/φ):
  thick rhombus → 2 thick + 1 thin
  thin  rhombus → 1 thick + 1 thin

Because φ is irrational, the thick-to-thin count can never settle into a whole-number ratio — and that irrationality is precisely what blocks the tiling from ever becoming periodic. This simulation generates P3 tilings via that deflation substitution, letting you step through successive inflation levels and watch the 5-fold symmetry hold at every scale.

Escaping the crystallographic restriction

For over a century, crystallographers relied on the crystallographic restriction theorem: a periodic arrangement in two or three dimensions can only possess rotational symmetries of order 1, 2, 3, 4 or 6 — five-fold symmetry was deemed impossible for any ordered solid, since regular pentagons cannot tile the plane without gaps. Because Penrose tilings are aperiodic rather than periodic, the theorem simply does not apply to them, and they happily display approximate five- and ten-fold rotational symmetry across the whole pattern.

From mathematics to a Nobel Prize

In 1982, physicist Dan Shechtman examined a rapidly cooled aluminium-manganese alloy and found electron diffraction images with sharp spots arranged in a clear ten-fold symmetry — long-range order that was supposedly forbidden. The result was so heretical he was reportedly asked to leave his research group. He was vindicated: the diffraction pattern was the signature of a quasicrystal, the three-dimensional physical analogue of a Penrose tiling. In 2011 Shechtman received the Nobel Prize in Chemistry, and the International Union of Crystallography broadened its definition of a crystal to any solid with a discrete diffraction pattern, periodic or not.

Frequently asked questions

Try it live

Everything above runs in your browser — open Penrose Tiling and step through deflation levels to watch the aperiodic pattern grow while its five-fold symmetry never breaks. Nothing is installed, nothing is uploaded.