Tile a floor with squares, hexagons, or equilateral triangles, and the pattern obediently repeats. Pick a translation by the right amount in the right direction, and the tiling looks identical to itself, indistinguishable from the original — that is what periodic means. The crystallographic restriction theorem says that any such repeating planar pattern can have rotational symmetry of order , , , or , but never order . Pentagonal symmetry is mathematically forbidden in a repeating wallpaper.
In 1974 the British mathematical physicist Roger Penrose published a pair of tile shapes that broke the rule by refusing to play the game. His tiles, combined with simple matching rules, cover the plane completely but never repeat. The resulting tilings are filled with patches that look fivefold-symmetric — pentagons and stars and decagons — and yet no translation maps the whole tiling onto itself. The restriction theorem does not apply because its hypothesis fails: the tilings are aperiodic, and the fivefold symmetry that is forbidden in periodic patterns is restored as a feature of these exotic non-repeating ones.
This article is about Penrose tilings — what they are, how they avoid repetition, what the deeper mathematics behind them looks like, and why the discovery of physical materials with the same kind of order won a Nobel Prize.
The two tiles, the five-fold star
Penrose actually discovered three pairs of tiles, all of which give aperiodic tilings. The most famous pair is the kite and dart, two quadrilaterals derived from a regular pentagon. A second pair is the thick and thin rhombi, both unit-edge rhombi with acute angles of and respectively. These two angles come from the regular pentagon ( is its central angle) and the regular decagon, which are the source of the fivefold flavour.
A famous local arrangement is the sun patch: five thick rhombi meet at a single central vertex, each contributing its acute angle. Since , the five rhombi fit perfectly, and the resulting shape has the exact fivefold rotational symmetry that the crystallographic restriction was supposed to forbid in periodic patterns.
The figure above is impossible in any periodic wallpaper pattern. Five thick rhombi sharing a vertex form a configuration with exact fivefold rotational symmetry, and no wallpaper-group symmetry can sustain that. Yet Penrose tilings produce it routinely; the sun patch occurs over and over in the full tiling, sometimes locally, sometimes within nested patterns of increasing scale.
The matching rules
The raw tile shapes alone are not enough to force aperiodicity. The same kites and darts can also be assembled into perfectly periodic patterns if you simply ignore the constraints. Matching rules are markings on the edges of the tiles that determine which edges may meet which. Penrose’s original rules used arrows: each tile edge had an arrow direction, and tiles could only be placed so that arrows on touching edges agree. A modern presentation often uses coloured arcs drawn on each tile, which must connect continuously across edges; the legal tilings are exactly those where the arcs form continuous unbroken curves.
The remarkable theorem — proved by Penrose and refined by others — is that every tiling of the plane by his two tile types that obeys the matching rules is aperiodic. There is no choice of placement at all that satisfies the rules and produces a periodic pattern. Aperiodicity is forced on the tiling not by avoiding it but by the rules themselves.
This is a stronger property than merely failing to repeat. A set of tiles is called aperiodic if all tilings using them are non-periodic, with no exceptions. Before Penrose, the smallest known aperiodic set, due to Raphael Robinson in 1971, had six tile types. Penrose dropped the count to four in 1973, then to two in 1974. The further fifty-year question — could a single tile type be aperiodic? — was answered “yes” only in 2023 by David Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss with their now-famous “hat” tile, which alone tiles the plane and only aperiodically.
Why aperiodicity is forced
The proof that Penrose tilings cannot be periodic is short and beautiful. Suppose, for contradiction, that a Penrose tiling were periodic, with a smallest translation that maps the tiling to itself. Count the ratio of the number of thick rhombi to thin rhombi within a single fundamental domain of . This ratio is a rational number, since it counts two finite quantities.
But there is an explicit calculation, due to Penrose, showing that the ratio of thick to thin rhombi in any Penrose tiling is exactly , the golden ratio. The golden ratio is an irrational number — it satisfies and has no integer reciprocal. An irrational ratio cannot equal a rational ratio. The contradiction shows no periodic tiling can satisfy the matching rules.
The golden ratio’s appearance is not a coincidence. The two tile types are related geometrically by golden-ratio scalings: a Penrose tiling can be inflated (each tile replaced by smaller pieces of the two types) or deflated (a group of tiles merged into a single larger tile), and the inflation/deflation operations introduce as a scale factor. The whole geometry of the tilings is governed by the golden ratio, which is also the central object in any honest discussion of the regular pentagon.
Inflation, decoration, and self-similarity
Penrose tilings have a beautiful self-similarity. Cover the plane with thick and thin rhombi in a legal way. Now subdivide each thick rhombus into two thick rhombi and one thin rhombus, scaled down by , in a specific pattern. The result is a finer Penrose tiling whose tile count is the original count multiplied by .
This inflation rule is one of the cleanest mechanisms for generating Penrose tilings explicitly: start with any small legal arrangement, repeatedly inflate, and you get a sequence of finite Penrose patches that converge in the limit to an infinite tiling of the whole plane. Different starting seeds produce different tilings, and there are in fact uncountably many distinct Penrose tilings up to translation — yet every two of them agree perfectly on any finite region (after a suitable translation). This last fact, called the local isomorphism theorem, is one of the most counter-intuitive features of aperiodic order: the global tilings differ wildly but locally are all the same.
Quasicrystals: the surprise from nature
The mathematical curiosity of Penrose tilings might have remained pure mathematics if not for an experimental discovery a decade later. In April 1982, the materials scientist Dan Shechtman was examining a rapidly cooled aluminium–manganese alloy under an electron microscope. The diffraction pattern showed sharp Bragg-like spots arranged in tenfold symmetry — a pattern that no crystal, by the crystallographic restriction theorem, could produce.
For two years Shechtman could find no journal willing to publish the result; the great chemist Linus Pauling famously remarked, “There are no quasicrystals, only quasi-scientists.” But the experiment was correct, and once the connection to Penrose-style aperiodic order was made, the field of quasicrystals was born. By 1991 the International Union of Crystallography had redefined the word “crystal” to include aperiodic structures, and in 2011 Shechtman received the Nobel Prize in Chemistry. Natural quasicrystals have since been found in meteorites that struck eastern Russia — they form under high-pressure conditions in some shocked meteoric materials, demonstrating that aperiodic order is not just a laboratory artefact but exists in the universe at large.
The atomic arrangement in a quasicrystal looks, to an X-ray, like a three-dimensional Penrose tiling. The mathematics of aperiodicity that Penrose worked out as a curiosity in the 1970s turned out to be the right framework for describing a class of physical materials that the rest of the world did not know existed.
What aperiodic order really teaches
Penrose tilings sit at the intersection of geometry, algebra, and the theory of dynamical systems. They show that the dichotomy “periodic versus random” is false: there is a third category, aperiodic order, in which a structure has all the long-range coherence of a crystal (a sharp diffraction pattern, well-defined repetition statistics, a clear notion of “structure”) without periodic repetition. Aperiodic order is a genuinely distinct kind of organisation, between the rigid symmetry of a crystal and the structural chaos of a random arrangement.
A second lesson is about the power of local rules. Two simple tile shapes with two simple matching rules, applied locally, force an infinite global structure with no repetition. There is no central coordinator, no overall blueprint — just the rules each tile follows when meeting its neighbour. Yet the rules produce, with absolute necessity, a pattern of staggering complexity and beauty. This is the geometric prototype of the broader phenomenon of emergent order: simple local interactions producing rich global structure, the same principle that underlies cellular automata, self-organising biological systems, and many condensed-matter phenomena.
A floor tiled with Penrose’s pieces never repeats. Every step you take across it brings you into territory that has never been seen before, and yet looks identical, on a small scale, to every other patch. There is, perhaps, no better physical demonstration in mathematics of the idea that infinite variation and strict structural coherence are not opposites but can live together in a single object. Roger Penrose discovered the geometric example in the early 1970s; nature, as it turns out, has been using a closely related arrangement to organise matter all along.
Frequently asked
Who discovered the first aperiodic tile sets?
Hao Wang asked in 1961 whether any finite set of tile shapes can tile the plane only aperiodically. He conjectured no such set existed. His student Robert Berger answered the conjecture in 1966 — with a set of 20,426 tile types. Subsequent mathematicians shrank the count: Berger himself to 104, then Donald Knuth to 92, Raphael Robinson to 6, and Roger Penrose to 4 (1973), then 2 (1974). In 2023, David Smith and collaborators discovered the 'hat' — a single shape that tiles the plane only aperiodically — answering a fifty-year-old question about whether one tile alone could do the job.
How does a Penrose tiling have fivefold symmetry if wallpaper groups forbid it?
The crystallographic restriction theorem only forbids fivefold symmetry in periodic patterns — those with a lattice of translation symmetries. Penrose tilings have no such lattice; there is no translation that maps the tiling exactly to itself. The fivefold symmetry of a Penrose tiling is statistical and local, appearing in finite patches and in the long-range diffraction pattern, but no exact global rotational symmetry exists either. The restriction theorem does not apply because its hypothesis — periodicity — fails.
What are the matching rules?
Penrose's two tile sets (kite-and-dart, or thick-and-thin rhombus) each come with markings or notches on their edges that constrain how the tiles can fit together. Without these rules, the same tiles can be used to make periodic tilings — but the rules force every legal tiling to be aperiodic. There are several equivalent ways to state the rules: coloured edges, arrows, or notched boundaries. Penrose's original 1974 paper used arrows; modern presentations often use coloured arcs drawn on the tiles whose continuity across edges enforces the constraint.
Are real materials with Penrose-like structure known?
Yes. In 1984 the chemist Dan Shechtman published a paper showing that a rapidly cooled aluminium–manganese alloy diffracted X-rays in a fivefold-symmetric pattern incompatible with any crystal — a quasicrystal. The discovery overturned a century of crystallography that had ruled out such structures and earned Shechtman the 2011 Nobel Prize in Chemistry. The atomic arrangement in quasicrystals turned out to have the same long-range order as Penrose tilings, with fivefold (and tenfold) symmetry. Quasicrystals have since been found in nature (in meteorites) and engineered for use in heat-resistant coatings, non-stick surfaces, and frying pans.