Penrose tilings

Two rhombi. One fat, with angles of $72°$ and $108°$; one thin, with $36°$ and $144°$. A rule about which edges may meet which. Between them they tile the plane completely — and they cannot tile it periodically, however they are arranged.

That is a strange combination of properties, and for most of the twentieth century it was thought to be impossible.

What aperiodic means #

The word is stronger than “not periodic”, and the distinction is the whole point.

A set of tiles is called aperiodic if it tiles the plane and every tiling it produces is non-periodic. Not “some arrangement fails to repeat” — any tile shape can be arranged badly — but “no arrangement repeats at all”.

That is a demanding property, and until 1964 nobody knew whether any set of tiles had it. Hao Wang had conjectured in 1961 that none did: that any set of tiles capable of covering the plane could cover it periodically. His student Robert Berger disproved it in 1964 with a set of 20,426 tiles, a number that came down over the following decade — Raphael Robinson reached six in 1971, and Roger Penrose reached two in 1974.

The two-tile case is where the subject became famous, and it is famous partly because the tiles are so simple that the impossibility of periodic arrangement is genuinely surprising.

The matching rules #

The two rhombi alone are not aperiodic. Left to themselves they will happily tile periodically — a fat rhombus is just a squashed square, and squashed squares tile like squares.

What forces aperiodicity is a matching rule: a constraint on which edges may be placed against which. In one common presentation, each edge carries an arrow, single or double, and edges may only meet when their markings agree.

Those markings are not decoration. They are the entire content of the construction, and without them the whole result evaporates. The tiles can also be given notched edges that enforce the same constraint physically — a jigsaw version — which is a useful reminder that the rule is a property of the shapes rather than an instruction imposed from outside.

The rules do two things at once. They forbid every periodic arrangement, and they leave enough freedom that the plane can still be covered. Neither alone would be interesting: a rule that forbade too much would make tiling impossible, and one that forbade too little would permit a repeat.

The inflation, which is the real generator #

Matching rules explain why a Penrose tiling cannot be periodic. They are a poor way to actually build one, because a patch assembled tile by tile according to local rules can reach a dead end — a position where no legal tile fits — and there is no local test that predicts it.

The construction that works is inflation. Start with a single tile. Cut each tile into smaller ones according to a fixed rule, rescale so the new tiles are the original size, and repeat. Every tiling produced this way is legal by construction, no dead end can occur, and the patch grows by a factor of $\varphi$ in linear size at each step.

Inflation is also what gives the tiling its most striking property. A Penrose tiling maps onto itself under a rescaling — enlarge by $\varphi$, redraw the tiles, and the same tiling reappears. That is a symmetry of a kind the four plane motions do not include, and it is what a quasicrystal has in place of a translation.

The theorem the tiling does not violate #

A reader arriving from the crystallographic restriction is entitled to be suspicious. Five-fold symmetry is impossible. This has five-fold symmetry. Something must be wrong.

Nothing is wrong. The restriction’s hypothesis is a lattice — a discrete group of translations with a shortest non-zero vector — and a Penrose tiling has no translations at all.

Every step of the descent proof fails without one. There is no shortest vector to contradict; the rotated copies of a translation are not translations, because there are none; and the sum construction has nothing to construct with. The theorem is not evaded or weakened. Its hypothesis is simply absent.

This is worth stating carefully because the popular account of quasicrystals frequently gets it backwards, describing them as a violation of a law of nature. They are nothing of the kind. They are a demonstration that one of the law’s hypotheses was being assumed silently.

What replaces periodicity #

A pattern with no translations could be arbitrary — a random scattering has no translations either. Penrose tilings are not arbitrary, and the property that makes them ordered has a name.

A tiling is repetitive if every finite patch that occurs in it occurs infinitely often, and moreover occurs within some bounded distance of every point. Penrose tilings are repetitive. Find a particular configuration of seven tiles somewhere, and an identical configuration is guaranteed within a fixed radius of anywhere else in the tiling.

That is a strong form of order. It means the tiling looks locally the same everywhere, in the precise sense that any finite window sees only a bounded number of distinct scenes, each of which recurs. What it does not mean is that any slide maps the whole tiling onto itself — the identical patches are in the wrong places relative to one another for that.

The difference between those two statements can be measured, and it is worth measuring because it is easy to say and hard to see. A translation symmetry would have to carry every vertex of the tiling to another vertex. So take the vertex nearest the middle of a patch, take the vector from it to each of the other vertices in turn — a translation symmetry has to be one of those, since it has to send that vertex somewhere — and for each candidate count how many of the patch’s vertices land on vertices.

The interesting part of that result is how close the best slide gets. A pattern that were merely disordered would score badly at every distance; this one very nearly repeats, and it does so further and further out. That near-miss is repetitivity showing up as a number — the patch reappears, endlessly, and never quite in register. A periodic pattern reaches one and stops looking; this one keeps climbing and never arrives.

Repetitivity is why the diffraction is sharp. Sharp reflections require long-range coherence, and repetitivity supplies it without supplying a lattice. The essay on order and periodicity takes that up.

Locally indistinguishable, globally distinct #

Here is the property that makes Penrose tilings genuinely strange, and it has no analogue among periodic patterns.

There are uncountably many distinct Penrose tilings — not seventeen, not two hundred and thirty, but a continuum of them. And any two of them are locally indistinguishable: every finite patch appearing in one appears in the other. No finite inspection, however large, can determine which tiling is being looked at.

For periodic patterns nothing like this happens. A finite patch of a p4m pattern, large enough, determines the whole pattern. Local information reconstructs the global object, because the global object is the local one repeated.

For a Penrose tiling, local information determines the whole family and never the individual. An observer with an arbitrarily large but finite view knows every rule the tiling obeys and cannot say which tiling it is.

Five-fold, and where exactly it is #

A small correction to the popular description, because “a Penrose tiling has five-fold symmetry” is usually said more loosely than it should be.

Most Penrose tilings have no exact five-fold rotational symmetry at all. Only two members of the uncountable family have a global five-fold centre — the ones grown from a symmetric seed — and the rest have none. What every Penrose tiling has is statistical five-fold symmetry: the distribution of tile orientations is exactly five-fold, and the diffraction pattern is exactly ten-fold.

That distinction matters for reading the literature. A quasicrystal is identified by its diffraction, not by finding a five-fold axis in its structure, and the diffraction symmetry is a statement about orientations rather than about any rotation mapping the material onto itself. Every tile edge in a Penrose tiling points in one of five directions, up to reversal; that orientational constraint is exact, holds for every member of the family, and is what produces the tenfold pattern.

Vertex types, which is what a tiling is made of #

The local structure can be enumerated, and doing so makes the constraint concrete.

In a Penrose rhombus tiling the matching rules permit exactly eight vertex configurations — eight ways that tiles may meet around a point. They have names: the sun, the star, the ace, the deuce, the jack, the queen, the king and the boat. Every vertex in every Penrose tiling is one of those eight, and no other arrangement of angles satisfying $360°$ is legal.

That is what a matching rule does when it is unpacked. Eight local configurations, propagating outward, and between them they make periodicity impossible while leaving the plane coverable.

The enumeration is also how a tiling is checked. Given a candidate patch, verifying it is a legal Penrose tiling is a matter of examining each vertex and asking which of the eight it is — a local, finite test, which is precisely what a matching rule promises.

Aperiodic tilings that are not Penrose’s #

The two rhombi are the famous case and they are not the only one, and the variety is worth knowing about because it shows which features are essential.

Robinson’s six tiles of 1971 use square tiles with notched edges, and they force aperiodicity by a hierarchical construction quite unlike Penrose’s — squares within squares within squares. No five-fold symmetry appears anywhere.

The pinwheel tiling, found by John Conway and Charles Radin in 1994, uses a single triangle and its mirror image, and its tiles appear in infinitely many orientations rather than five. Its diffraction is correspondingly circular rather than spotty.

The hat, announced in 2023 by David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss, is a single tile that forces aperiodicity when reflections are permitted — the long-sought “einstein”, from ein Stein, one stone. A follow-up tile, the spectre, does it without needing reflections.

What that variety establishes is that aperiodicity is not a five-fold phenomenon. Five-fold symmetry is the reason Penrose’s tiles connect to crystallography, but the underlying property — a tile set that covers the plane and forbids every repeat — is more general and turns up in constructions with no forbidden symmetry in them at all.

It also establishes something about what a picture can settle, which is worth saying plainly because this page is mostly pictures. Set a patch of a periodic pattern beside a patch of an aperiodic one, at the same density and the same window size, and nothing in either one announces which is which. Both are completely determined; neither is random; the pinwheel’s tiles even appear in infinitely many orientations, so a reader looking for a forbidden symmetry would find no symmetry at all and still be looking at a tiling as rigid as a crystal. The difference between a repeat and a recurrence is a statement about the infinite object, and the only way to get at it from a finite window is to measure — which is what the slide scan above does, and why it is on the page rather than a pair of pictures inviting a comparison the eye cannot make.

The ratio, measured #

The two tiles do not appear in equal numbers, and the ratio between them is the tiling’s most quoted property.

In the limit of an infinite tiling, the number of thick tiles to thin ones approaches the golden ratio, $\varphi = (1+\sqrt5)/2 \approx 1.618$. Not approximately in the sense of a rough fit — the ratio converges to it exactly, and it converges because each round of inflation multiplies the tile counts by a matrix whose dominant eigenvalue is $\varphi^2$.

That the ratio is irrational is not incidental. It is the reason the tiling cannot be periodic. A periodic tiling has a unit cell containing a whole number of each tile, so the ratio of tile counts in a periodic tiling is necessarily rational. An irrational ratio is therefore a proof of aperiodicity, and it is the cleanest one available.

The number that made five-fold impossible #

There is a connection between this essay and the impossibility proof that is too neat to leave unremarked.

The descent argument rules out five-fold periodicity by constructing a vector of length $2\cos 72° \approx 0.618$ times the shortest — and $0.618\ldots$ is $1/\varphi$.

So the number that makes five-fold periodicity impossible is the reciprocal of the number that governs five-fold aperiodic order. That is not a coincidence. The golden ratio is the fundamental unit of the algebra generated by five-fold rotation, so every quantity built out of $72°$ angles is an expression in it. Because $\varphi$ is irrational, five-fold periodicity fails; because it is a quadratic irrational, the tiling has an inflation rule and is therefore self-similar rather than merely non-repeating.

The impossibility and the replacement come out of the same algebraic fact, seen from two sides.

What this site measures, and what it does not #

An honest statement of scope, because the machinery that verifies every other figure here does not apply to this one.

The round trip that certifies periodic patterns needs a lattice: it works in coordinates along the repeat vectors, where operations are integer matrices and translations are small rationals. A Penrose tiling has no repeat vectors, so there is nothing to choose as coordinates, no finite holohedry to enumerate, and no exact comparison to make.

So these figures are constructed and measured rather than generated and proved. The tilings are built by inflation from a seed, which guarantees legality by construction; the tile counts are counted; and the ratio is reported as a measurement converging on $\varphi$. Where a number appears in an aperiodic figure on this site, it came from counting rather than from a theorem.

That is a weaker claim than the periodic figures make, and it is stated rather than glossed. The alternative — implying that the integer machinery decided something about an aperiodic tiling — would undermine every claim on the site.

Penrose, Kepler, and the medieval craftsmen #

The history has three layers and each is more surprising than the last.

Roger Penrose found the two-tile set in 1974, working, by his own account, as recreation. He had been interested in tiling problems since school and came at it from mathematical physics rather than crystallography. The tiles were popularised by Martin Gardner in Scientific American in 1977, which is how most people met them.

Johannes Kepler got there first, partly. His Harmonices Mundi of 1619 contains a figure — plate Aa — showing pentagons, pentagrams and decagons in an arrangement that cannot be completed periodically, and Kepler’s text remarks that it may be continued indefinitely. He did not have the concept of an aperiodic tile set, but he had noticed the phenomenon.

And in 2007 Peter Lu and Paul Steinhardt showed that the girih tilings of medieval Islamic architecture — in particular the Darb-i Imam shrine at Isfahan, completed in 1453 — are built from a set of five decorated tiles that produce patterns with near-perfect Penrose structure, including the correct inflation relationship. Whether the craftsmen understood the aperiodicity is unknowable and probably unlikely. That they arrived at the construction five centuries early is not in doubt.

Where the ladder goes next #

The mechanism that generates these tilings is inflation, which is also where the golden ratio comes from and what a quasicrystal has in place of a repeat.

The physical payoff is order without periodicity — what Shechtman measured, why it was disbelieved, and how the definition of “crystal” was rewritten in 1992 to accommodate it.

And the theorem that all of this politely does not violate is the crystallographic restriction, whose hypothesis is exactly the thing a Penrose tiling lacks.

What the pictures here cannot show. Every figure on this page is a finite patch, and aperiodicity is a property of the infinite tiling. A finite patch of a Penrose tiling is, by itself, entirely compatible with being part of a periodic pattern — the claim that no periodic continuation exists is a claim about the matching rules, not about anything visible in the drawing.

Three tile sets, one tiling #

The phrase Penrose tiling names three different pairs of tiles, and a reader meeting two of them can reasonably wonder which is the real one. The answer is that the question does not arise, and why it does not is worth stating.

The rhombs are the pair this essay uses: a fat rhombus and a thin one, with matching rules on their edges.

The kites and darts are the other famous pair — a convex quadrilateral and a re-entrant one, cut from a rhombus along a line dividing it in the golden ratio. They are the set most often drawn in popular accounts and they carry their own matching rules.

And the original set is Penrose’s first, from 1974: six tiles including pentagons, a five-pointed star, a boat and a diamond, with rules about how the pentagons may meet.

All three produce the same tilings, in a precise sense. Given a tiling by one set, the tiling by another can be constructed from it by a local rule — look at a bounded neighbourhood, and place the new tiles accordingly — and the construction runs both ways. Two tilings related that way are called mutually locally derivable, and it is the right notion of sameness for objects with no lattice to compare.

So the three sets are three descriptions of one object, exactly as three symbols can describe one space group and several cells one lattice. The tiles are the convention and the tiling is the thing, and every property this essay reports — the inflation factor, the repetitivity, the vertex count, the diffraction — is a property of the tiling and takes the same value under each description.

The vertex count is the one place the descriptions visibly differ. Eight vertex types for the rhombs, seven for the kites and darts, because the two sets cut the same arrangement in different places — and the discrepancy is a fact about the cutting rather than about the pattern.