Penrose Tiling Generator
Robinson-triangle deflation builds an infinite aperiodic mosaic with five-fold symmetry — where the golden ratio φ≈1.618 governs every proportion.
💠 Aperiodic Order
A Penrose tiling covers the infinite plane with no gaps and no overlaps, yet it never repeats by any translational shift. Any finite patch does appear infinitely often — but always in slightly different contexts.
🌟 Five-fold Symmetry
The tiling is invariant under 72° rotations around certain special points — a five-fold symmetry that classical periodic crystals forbid. Crystallographic restriction allows only 2-, 3-, 4- and 6-fold symmetries for periodic lattices.
𝜒 Golden Ratio φ
Every length, frequency ratio and tile-count ratio in the Penrose tiling is a power of φ=(1+√5)/2≈1.618. The ratio of fat to thin rhombuses in any large patch converges to φ.
🔬 Quasicrystals
Dan Shechtman discovered aperiodic Al–Mn alloys in 1982; Roger Penrose had described the 2D tiling in 1974. The 3D analogue explains the icosahedral Bragg diffraction peaks in real quasicrystals. Shechtman received the 2011 Nobel Prize in Chemistry.
What Is the Penrose P3 Tiling?
Roger Penrose discovered in 1974 that the plane can be tiled using just two shapes — a fat rhombus (angles 72°/108°) and a thin rhombus (angles 36°/144°) — provided tiles obey matching rules (marked by coloured arcs or arrows on physical puzzle tiles). The result is an aperiodic tiling: infinite yet never periodic.
The two rhombus shapes tile the plane in uncountably many distinct ways; all of them are locally isomorphic (every finite patch appears in every valid Penrose tiling). This property is called local isomorphism and implies that no finite examination can distinguish one tiling from another.
Robinson-Triangle Deflation
Each rhombus is split diagonally into two Robinson half-rhombus triangles. There are two types:
Type 1 (thin half, 108° apex A): Thin triangle with sides AB=AC, base BC=φ·side
Deflation replaces each triangle by smaller copies, scaled by 1/φ:
Type 1 → 1 fat + 1 thin (2 children total)
After n deflations the number of triangles grows as φn+1. Depth 7 yields ≈750k half-triangles (≈375k full rhombuses).
Golden Ratio in Numbers
| Depth | Total half-triangles | Fat rhombuses | Thin rhombuses | Ratio fat:thin |
|---|---|---|---|---|
| 1 | 30 | 10 | 5 | 2.000 |
| 2 | 70 | 25 | 10 | 2.500 |
| 3 | 170 | 60 | 25 | 2.400 |
| 4 | 410 | 145 | 60 | 2.417 |
| 5 | 990 | 350 | 145 | 2.414 → φ+1 |
| 6 | 2 390 | 845 | 350 | → φ²/φ = φ |
| 7 | 5 770 | 2 040 | 845 | 2.414 ≈ 1+φ¹ |
(Starting from a 10-triangle sun; visible tile count higher at interactive depths due to pre-image sampling.)
Why No Periodicity?
Suppose a Penrose tiling had a translational period v. Scaling by φ maps any Penrose tiling to another valid Penrose tiling, so the scaled period φv would also be a period. Iterating, φnv would be a period for every integer n. As φ>1, these grow without bound, contradicting the assumption that the tiling has any finite patch that uniquely places the period. More rigorously: because φ is irrational, no power of φ is rational, so no lattice translation can simultaneously commute with both the deflation and its inverse (inflation). The tiling is therefore provably non-periodic.
FAQ
What makes Penrose tiling special?
Penrose tilings are aperiodic: they cover the full plane with no gaps or overlaps, yet never repeat by any translational shift. Despite this, they have perfect long-range quasicrystalline order — five-fold rotational symmetry (forbidden in classical crystals) and all distances governed by the golden ratio.
Why does the golden ratio appear?
The rhombus angles (multiples of 36°=π/5) force diagonal ratios of φ:1. Deflation replaces each tile with φ² scaled copies, and the fat-to-thin ratio converges to φ — ensuring aperiodicity because φ is irrational and so no periodic translation lattice exists.
How does Robinson-triangle deflation work?
Each rhombus is split into two Robinson half-triangles. A fat half deflates into 2 fat + 1 thin children; a thin half into 1 fat + 1 thin. After n iterations the number of tiles multiplies by φ²≈2.618, producing arbitrarily fine aperiodic tilings.
Do Penrose tilings appear in nature?
Yes. In 1982 Dan Shechtman found Al–Mn alloys with icosahedral Bragg diffraction patterns, explained by a 3D quasicrystalline structure analogous to Penrose tiling. He received the 2011 Nobel Prize in Chemistry. Quasicrystals have since been found in meteorites and engineered for photonic and thermal applications.