Penrose Tiling: A Pattern That Never Repeats

Two tiles and a single rule

A Penrose tiling covers the entire Euclidean plane, with no gaps and no overlaps, using copies of just two shapes. Penrose gave several equivalent sets. The most famous is the kite and dart (his 1974 ‘P2’ tiling): the kite is a convex quadrilateral with interior angles 72°, 72°, 72°, 144°, and the dart is a non-convex ‘arrowhead’ with angles 36°, 72°, 36°, 216°. An equally common version, ‘P3’, uses two rhombi of equal side length — a thick rhombus (angles 72° and 108°) and a thin rhombus (36° and 144°).

The subtlety hides in one word: only. These same shapes, left to themselves, will happily tile the plane periodically — you can lay them out in a dull repeating strip. What makes a tiling ‘Penrose’ is a set of local matching rules: each edge is decorated with arrows, colours, or interlocking bumps and notches, and two tiles may meet only when their markings agree. With the rules enforced, something remarkable happens — the tiles can still cover the whole plane, but they can never do so periodically. A finite set of tiles with this property is called an aperiodic set, and Penrose’s two tiles are the most famous example. (The first aperiodic sets, found by Berger in 1966, needed over 20,000 distinct tiles; Penrose got the count down to two.)

The golden ratio is built into the shapes

Every length and angle in a Penrose tiling is governed by the golden ratio, φ = (1 + √5)/2 ≈ 1.6180339887, the number satisfying φ² = φ + 1 and 1/φ = φ − 1. Both tiles are assembled from two ‘Robinson triangles’: the acute golden triangle (angles 36°–72°–72°) and the obtuse golden gnomon (36°–36°–108°). In each, the ratio of the long side to the short side is exactly φ. Glue two acute triangles along their long edge and you get a kite; glue two obtuse ones and you get a dart.

The five-fold flavour comes straight out of trigonometry: 2·cos 36° = φ and 2·cos 72° = φ − 1. Because 36° = π/5 and 72° = 2π/5 are the diagonal angles of a regular pentagon, the golden ratio and five-fold symmetry are two faces of the same object. In the rhombus version the link is even cleaner: the thick rhombus has area sin 72° ≈ 0.951 and the thin one sin 36° ≈ 0.588, so their area ratio is sin 72° / sin 36° = 2·cos 36° = φ. As we are about to see, that irrational ratio is not decoration — it is the reason the pattern can never repeat.

Inflation and deflation: a self-similar hierarchy

The engine that builds a Penrose tiling is inflation and deflation — a ‘substitution rule’. Deflation cuts each tile into smaller kites and darts: a kite splits into two smaller kites and two half-darts, a dart into one kite and two half-darts, with every new tile scaled down by a factor of 1/φ. Repeat, and the tiles shrink toward zero while their number explodes. Inflation is the reverse operation — grouping tiles into larger ‘super-tiles’ scaled up by φ. Because each tile sits inside a unique larger tile, which sits inside a larger one still, a Penrose tiling has an infinite hierarchical, self-similar structure: it looks like a rescaled copy of itself at every power of φ.

This substitution also proves the tiling exists and fills the plane. Encode ‘how many of each tile a tile deflates into’ as a substitution matrix; for the two Robinson triangles it is [[2, 1], [1, 1]] (the square of the Fibonacci matrix), whose largest (Perron–Frobenius) eigenvalue is exactly φ². Iterating the substitution grows lengths by φ and areas — hence tile counts — by φ² ≈ 2.618 per step, and the leading eigenvector fixes the relative frequencies of the tiles. In the limit the number of kites divided by the number of darts converges to φ (equivalently, thick rhombi to thin rhombi tends to φ). Hold on to that number.

Why it can never repeat

Here is a complete, elementary proof that no Penrose tiling is periodic. Suppose one were. Then it would have a fundamental domain — a finite ‘unit cell’ that repeats by translation — containing, say, p kites and q darts. Every region of the tiling would then contain kites and darts in the whole-number ratio p : q, so the overall frequency ratio would be the rational number p/q. But we just computed that in a Penrose tiling the kite-to-dart ratio equals φ, which is irrational. A rational number can never equal an irrational one, so no fundamental domain can exist. The tiling cannot repeat. Full stop.

The five-fold symmetry runs into a second, independent wall: the crystallographic restriction theorem. Any pattern that does repeat periodically sits on a lattice, and an n-fold rotation must map that lattice to itself. Written in a lattice basis, the rotation becomes an integer matrix, so its trace — which for a 2-D rotation equals 2·cos(2π/n) — must be a whole number. Solving 2·cos(2π/n) ∈ {−2, −1, 0, 1, 2} leaves only n = 1, 2, 3, 4, 6. Five-fold symmetry is impossible for any periodic tiling because 2·cos 72° = 0.618… is not an integer. Penrose tilings display gorgeous local five- and ten-fold symmetry precisely because they refuse to be periodic — they slip through the loophole the theorem leaves open.

Everything local repeats; the whole never does

Non-repetition is not disorder. Penrose tilings are repetitive (or ‘locally isomorphic’): every finite patch that appears anywhere reappears infinitely often, and — by a theorem of John Conway — a patch of diameter d recurs within a distance of roughly two diameters of any point you choose. So a local observer walking the tiling meets the same neighbourhoods over and over; only a global, God’s-eye comparison could reveal that the whole never lines up. In fact, only seven distinct vertex configurations are ever legal — Conway nicknamed them the sun, star, ace, deuce, jack, queen and king.

Two further facts sharpen the paradox. First, there are uncountably many (2^ℵ₀, a full continuum) genuinely different Penrose tilings, yet they are locally indistinguishable: any bounded patch you find in one occurs in every other, so no finite inspection can ever tell two of them apart. Second, the hidden bookkeeping becomes visible as Ammann bars — five families of straight lines running across the tiles. Their spacings take only two values, a long L and a short S with L/S = φ, arranged in the Fibonacci word (…LSLLSLSL…), a one-dimensional sequence that is itself the simplest quasiperiodic pattern. And to preempt a common guess: a Penrose tiling is not a fractal. It fills genuine two-dimensional area (dimension exactly 2); its self-similarity is the hierarchical φ-scaling of the substitution, not a fractional Hausdorff dimension.

Cut-and-project: a shadow of a 5-D lattice

Where does the five-fold, golden-ratio structure ultimately come from? The deepest answer is the cut-and-project method, worked out for Penrose tilings by N. G. de Bruijn in 1981. Take the integer lattice ℤ⁵ in five-dimensional space and split that space into a two-dimensional ‘physical’ plane and a three-dimensional ‘internal’ space, tilted at an irrational angle fixed by φ. Select the lattice points whose internal-space shadow lands inside a fixed bounded window (an acceptance region), then project those points onto the physical plane: the pattern they make is exactly a Penrose tiling. The five appears because ℤ⁵ carries a natural five-fold symmetry (cyclically permuting the coordinate axes); the golden ratio appears because the projection slope is irrational.

Equivalently, de Bruijn’s pentagrid builds the tiling directly in the plane from five families of equally spaced parallel lines set at 72° to one another; every intersection becomes a rhombus. This viewpoint explains a startling physical fact: although a Penrose tiling has no periodicity, its diffraction pattern is a set of sharp spots — a pure-point spectrum — arranged with ten-fold symmetry, the signature of genuine long-range order. Truly disordered materials such as glass produce only a diffuse blur; a Penrose pattern produces crisp Bragg peaks, just like a crystal.

From a puzzle to Nobel Prizes and a single tile

That last fact stopped being a curiosity in 1982, when materials scientist Dan Shechtman, studying a rapidly cooled aluminium–manganese alloy, recorded an electron-diffraction pattern with ten sharp spots in a perfect decagon — five-fold symmetry that ‘could not exist’. His result drew ridicule; Linus Pauling famously scoffed that there were ‘no quasicrystals, only quasi-scientists’. Yet Shechtman was right: he had found a quasicrystal, real matter with long-range aperiodic order, and the mathematics describing it was Penrose’s. The three-dimensional analogue tiles space with two golden rhombohedra (Ammann’s tiles) and captures the icosahedral symmetry of real quasicrystals. The term ‘quasicrystal’ was coined by Levine and Steinhardt in 1984; Shechtman received the 2011 Nobel Prize in Chemistry.

The story is still open. For nearly fifty years mathematicians asked whether a single tile could force aperiodicity — the ‘einstein problem’ (German ein Stein, ‘one stone’, not the physicist). In 2023 the hobbyist David Smith, with Craig Kaplan, Joseph Myers and Chaim Goodman-Strauss, found the ‘hat’, a 13-sided polygon that tiles the plane only aperiodically, and months later the ‘spectre’, which does so without even needing mirror-image copies. Penrose’s playful puzzle — he once sued a company for embossing his pattern onto quilted toilet paper — thus runs from a 1970s doodle to Nobel-winning chemistry and a problem solved only last year.

Three ways to fill the plane: an ordinary repeating crystal, a Penrose (quasicrystalline) tiling, and a truly disordered glass.

Property	Periodic crystal	Penrose tiling (quasicrystal)	Amorphous glass
Translational symmetry	Yes — repeats on a lattice	None — never lines up with itself	None
Rotational symmetry	Only 1, 2, 3, 4, 6-fold	Local 5-fold and 10-fold	None (random)
Long-range order	Yes	Yes (quasiperiodic)	No
Diffraction pattern	Sharp Bragg peaks	Sharp Bragg peaks, 10-fold symmetry	Diffuse halo
Patch recurrence	Exact periodic repeat	Every patch recurs, whole never repeats	No recurrence
Textbook example	Square or hexagon tiling	Kite & dart / thick & thin rhomb	Window glass