Penrose Tiling — P3
Two rhombus shapes that tile the plane without ever repeating — a non-periodic tiling with fivefold symmetry.
Two Tiles
A P3 Penrose tiling uses two rhombus shapes: a thick rhombus (angles 72° and 108°) and a thin rhombus (angles 36° and 144°). The tiles must be assembled following matching rules — arrows on the edges that must be aligned.
Thick rhombus: angles 72°/108°, ratio 1:φ sides
Thin rhombus: angles 36°/144°, ratio 1:φ sides
where φ = (1 + √5) / 2 is the golden ratio.
A central decagonal cluster: ten thick rhombi arranged around a central point with perfect fivefold symmetry. This is the seed for the inflation construction.
Non-Periodicity
No Penrose tiling is periodic. You cannot pick up the pattern and slide it to match itself exactly. Yet any finite patch appears infinitely often throughout the tiling — the pattern is quasiperiodic.
The tiling extended outward. Fivefold symmetry is visible at large scale — a symmetry impossible in any periodic crystal. In 1982, crystallographer Dan Shechtman discovered quasicrystals with exactly this diffraction symmetry, eventually winning the Nobel Prize in Chemistry.