Penrose Tiling Generator
Generate Roger Penrose's famous P3 rhombic tiling by repeatedly deflating Robinson triangles. Walk through the subdivision depth, swap palettes, and confirm in real time that the ratio of thick to thin tiles converges to the golden ratio φ.
Starting shape
What makes a tiling "Penrose"
A Penrose tiling covers the plane with copies of two shapes and never repeats. Translate the pattern by any vector and it does not line up with itself — yet zoom out and the pattern is locally indistinguishable everywhere. It has long-range order without periodicity, which is exactly the property the world thought was impossible until physicist Dan Shechtman discovered real quasicrystals in 1982.
Roger Penrose published several inequivalent versions; this generator builds the P3 tiling from two rhombi:
- Thick rhombus — angles 72° and 108°. Same side length as the thin one.
- Thin rhombus — angles 36° and 144°. The narrow lozenge.
You never see the two arranged in a repeating brick wall. The matching rules force a non-repeating global structure no matter how cleverly you try to lay them.
How the tool generates it: deflation
The cleanest way to build a guaranteed-valid Penrose patch is the deflation (or inflation) algorithm. Each rhombus is split along its long diagonal into two Robinson triangles, then every triangle is subdivided into smaller Robinson triangles by a fixed rule:
Acute (thin half, 36°-72°-72°)
One thin half + one thick half.
Obtuse (thick half, 108°-36°-36°)
One thin half + two thick halves.
The dividing points are placed using the golden ratio φ = (1 + √5) / 2 ≈ 1.61803. Specifically, every edge cut happens at distance L / φ from a chosen endpoint, where L is the edge length. Repeating the rule n times multiplies the tile count by roughly φ² each step.
The golden-ratio fingerprint
In an infinite Penrose tiling the ratio of thick rhombi to thin rhombi is exactly φ. Try this in the live generator: bump the subdivision depth and watch the "Thick / Thin" readout converge to 1.6180… as the count grows. The denser the patch, the closer the ratio.
The same φ shows up in the ratio of long-to-short distances in the pattern, in the lengths of the two rhombi's diagonals, and in the spacing of the tiles' characteristic spectrum — which is exactly the diffraction signature physicists found in metallic quasicrystals like Al72Ni20Co8.
A short history
- 1961 — Hao Wang asks: is there a set of tiles that only tile the plane aperiodically? He expects the answer to be no.
- 1966 — Robert Berger finds a set of 20,426 tiles that does the job. Later refinements bring it down to 104, then 13.
- 1974 — Roger Penrose finds a set of just two tiles (the kite and the dart, and equivalently the two rhombi). Aperiodic tilings become beautiful.
- 1982 — Dan Shechtman discovers metal alloys whose X-ray diffraction shows 10-fold symmetry — forbidden for periodic crystals. The community dismisses him for years; he wins the 2011 Nobel Prize in Chemistry.
- 2023 — David Smith's "hat" tile becomes the first single aperiodic monotile (with reflections), followed by the "spectre" which needs no reflections.
Things worth trying
Crank depth from 0 to 7. At depth 0 you see the starting wheel; by depth 7 the generator is drawing tens of thousands of tiles. Watch the Thick/Thin counter glide toward φ.
Switch starting shape. The Sun and the Single Rhombus produce wildly different global patterns even though they obey the same local rules — a hint that the tiling is not unique.
Hide outlines for art. Turn off outlines, pick the Mono palette, download the PNG and you have a wallpaper. The geometry never repeats — your eye never quite settles.