Topic · Geometry

Penrose Tilings

Two shapes that cover the plane and never repeat.

A kite and a dart. Two simple tiles, four matching rules, and a plane that fills forever without any finite patch ever recurring. Roger Penrose described the construction in 1974; nature copied it in 1982 with quasicrystals. The pattern hides the golden ratio in plain sight.

Discover the story → Open the Explorer

First encounter

Two tiles. Cover the plane. Never repeat.

The big idea

Take two simple shapes — a kite and a dart, cut from a single rhombus. Lay them down side by side, respecting a few simple matching rules on the edges. They cover the whole infinite plane perfectly, like normal tiles. But here is the twist: no finite chunk of the pattern ever appears twice in the same orientation. Anywhere. Ever. Roger Penrose found this in 1974, and it broke a decades-old question.

A concrete example

Count the tiles in any large patch and you find the same number waiting: kites over darts always converges to φ = (1+√5)/2 ≈ 1.618. There is also a rule called inflation: every tiling has a unique parent tiling at the larger scale and a unique child tiling at the smaller — and each step scales lengths by φ exactly. The golden ratio is not decoration; it is the structure.

kites / darts → φ

each deflation scales by 1/φ

Why it matters

For 20 years mathematicians wondered: could a finite tile set force only aperiodic tilings? Berger answered yes with 20,426 tiles in 1966; Penrose dropped it to two in 1974. Then in 1982 Dan Shechtman fired electrons at an aluminium-manganese alloy and saw five-fold diffraction patterns — impossible under classical crystallography. Penrose tilings were the existing piece of paper-mathematics that proved nature was allowed to do this. Shechtman got the Nobel in 2011.

Below: grow a tiling by inflation, then watch φ emerge from the Fibonacci spiral.

Section 01 · The crystallographic restriction

Five-fold symmetry is forbidden in classical crystals

A classical crystal is a periodic tiling — a wallpaper pattern. A theorem from the 19th century says that any wallpaper repeating in two independent directions can only have 2-, 3-, 4-, or 6-fold rotational symmetry. Five-fold is impossible: pentagons cannot tile the plane without gaps. So if you ever saw a material with sharp diffraction spots arranged with 5-fold symmetry, the rulebook said you were hallucinating. The Penrose tiling sidesteps the theorem by refusing to be periodic in the first place — it can have local five-fold order at every scale precisely because it never closes into a repeating cell.

Crystallographic restriction

allowed: 2 · 3 · 4 · 6 forbidden: 5 · 7 · 8 · …

Only 2-, 3-, 4-, and 6-fold rotations are compatible with a 2-D lattice. Five-fold is the simplest forbidden order, and the one Penrose tilings deliver anyway.

Section 02 · From 20,426 tiles to one

Wang → Berger → Penrose → einstein

In 1961 Hao Wang asked: is there an algorithm that decides whether a given set of tiles can tile the plane? He noticed his question reduces to whether any aperiodic tile set exists. His student Robert Berger answered in 1966: yes, and he gave one — a Frankenstein set of 20,426 Wang tiles. Other mathematicians whittled the count down through the late 1960s. Then in 1974 Penrose found just two tiles. For 49 years that was the record. In 2023 David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss announced the einstein — a single tile (one polygon, no matching rules!) that tiles the plane only aperiodically. They followed it months later with the spectre, which doesn't even need reflections. The 60-year hunt ended with one shape.

The tile-count race

year	who	tiles
1961	Wang · question posed	?
1966	Berger · first answer	20,426
1968	Knuth	92
1971	Robinson	6
1974	Penrose · kite + dart	2
2023	Smith / einstein	1
2023	Smith / spectre	1 (no reflections)

Section 03 · Kite and dart with matching rules

The matching rules force aperiodicity

From two golden triangles — a 72-72-36 acute triangle and a 108-36-36 obtuse triangle — Penrose assembles the kite and the dart. (The P3 set of thin and thick rhombs is a separate, dual family — not produced by halving the same rhombus.) With no further restrictions you could tile the plane periodically using just kites and darts. The trick is the colored arcs Conway drew on each tile's edges: one red, one green, both crossing the kite and dart in specific places. The rule is then simply: where two tiles meet, their arcs must continue. With this single constraint every legal tiling becomes aperiodic. The arcs grow into never-closing curves that wander the whole plane.

Interactive · the deflation rule

Start with a sun, deflate by 1/φ, watch a tiling appear

This grows the P3 rhombus form, the thin-and-thick sibling of the kite and dart, from the canonical sun seed: a five-fold star of ten half-rhombi around the origin. Each deflation step subdivides every tile by Penrose's rules. Slide the depth up: at level 1 you can still see the seed; by level 6 you are looking at a few thousand half-tiles and the thick/thin ratio is already several digits of φ.

Interactive · deflate the tiling

Deflation depth

4340 half-tiles

▰ thick rhombi105

▰ thin rhombi65

thick / thin1.6154

φ ≈1.6180

Each deflation step subdivides every tile into pieces 1/φ times smaller. Thick rhombi outnumber thin ones, so the thick-to-thin count ratio walks toward φ ≈ 1.618 as depth grows. Five-fold symmetry is preserved exactly at every step.

Section 04 · Inflation and deflation

Every tiling has a unique parent and unique children

Penrose's deepest move was the deflation rule. Take any kite-and-dart tiling and subdivide every tile by a fixed recipe: each half-kite becomes two smaller half-kites and one half-dart, each half-dart becomes one half-kite and one half-dart. The pieces reassemble into a Penrose tiling at 1/φ the scale. Run the recipe backwards (inflation) and neighbouring tiles merge into a unique parent tiling at φ times the scale. The parent is unique: there is only one way to coarsen a Penrose tiling into a Penrose tiling. That is what locks the pattern into aperiodicity, because any periodic tiling would eventually inflate into itself, yet the φ-scaling is irrational and never closes.

Inflation

length_n+1 = φ · length_n

Inflation grows the tiling by exactly φ each step. Because φ is irrational, no inflated copy ever lines up with itself, and that is the formal reason the tiling cannot be periodic.

Section 05 · The golden ratio everywhere

Every measurement ratio contains φ

Once you start looking, φ = (1+√5)/2 is everywhere in the tiling. The number of kites divided by the number of darts in any large patch converges to φ. The side-length ratios inside the underlying Robinson triangles are 1 : φ. The inflation rule scales every length by exactly φ. The kite covers exactly φ times the area of the dart, and every tile is built from just two edge lengths whose ratio is φ. φ-scaling is also why you can prove the tiling is aperiodic without any geometry at all: a periodic tiling would force kites/darts to be a rational number, and φ is the most irrational number there is — its continued-fraction expansion is [1; 1, 1, 1, …], the slowest convergent in all of mathematics.

Interactive · why φ shows up everywhere

1, 1, 2, 3, 5, 8, 13 … and the spiral that converges to φ

Place a 1×1 square. Next to it, another 1×1. Then a 2×2 along their joined edge. Then a 3×3, then 5×5, then 8×8 — each new square's side length is the sum of the previous two. Inside each square draw a quarter-circle. The arcs join into a logarithmic spiral whose growth factor is φ. Slide the level up to watch the ratio of consecutive Fibonacci numbers crash into φ exponentially fast.

Interactive · the golden ratio

φ = (1 + √5) / 2

1.61803…

Fibonacci levels

7F = 13

Consecutive Fibonacci ratio

13 / 81.625000

Δ from φ : 6.97e-3

Each new square has the side length of the next Fibonacci number, and rotates around the previous one in 90° steps. The quarter-circle arcs join into the golden spiral — the same curve hidden inside every Penrose patch.

Section 06 · Quasicrystals — the picture became matter

Shechtman, 1982 · Nobel, 2011

On April 8, 1982, Dan Shechtman at NIST shot electrons through a rapidly quenched aluminium-manganese alloy and recorded a diffraction pattern with sharp 10-fold (and therefore 5-fold) symmetry. By every textbook rule it should not have existed. He spent two years convincing colleagues he had not made a mistake; Linus Pauling spent the next decade insisting publicly that 'there is no such thing as quasicrystals, only quasi-scientists.' Shechtman published in 1984. The Penrose tiling was the existing piece of mathematics that proved the diffraction pattern was consistent — long-range order without periodicity, five-fold symmetry allowed by aperiodicity. Today quasicrystals are an entire branch of materials science, used in non-stick coatings and even found in a meteorite from Russia. Shechtman received the 2011 Nobel Prize in Chemistry. A drawing on paper preceded a piece of matter by eight years.

Timeline · paper → matter → Nobel

1974Penrose publishes the kite and dart aperiodic tiling.
1982Shechtman records 5-fold electron diffraction in Al-Mn.
1984Shechtman et al. publish; the term quasicrystal is coined.
2009Natural quasicrystal found in a Russian meteorite (icosahedrite).
2011Shechtman receives the Nobel Prize in Chemistry.
2023Smith, Goodman-Strauss et al. announce the einstein and the spectre.

Take it further

Open the Explorer.

The Explorer lets you push the deflation deeper, toggle between the full rhombi and the Robinson half-triangles that build them, switch on Conway's matching markers, and rotate the seed. Every claim on this page becomes a knob there.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Quasicrystals

Dan Shechtman's 1982 discovery of metallic quasicrystals (Nobel 2011) was understood through Penrose tilings — the same five-fold mathematics governs both.

Architecture

The Storey Hall facade in Melbourne and several Islamic-mathematical patterns at Topkapı Palace use Penrose-style aperiodic geometry.

Materials science

Quasicrystalline coatings (e.g. on Sjöbo non-stick pans) are commercial today, exploiting Penrose-tiling-style atomic arrangements.

Cryptography

Recently proposed pseudorandom-number generators use aperiodic tiling sequences for low-discrepancy sampling.