Crystallography & Symmetry
patterns derived from their groups
Order without repetition

Matching rules, and what actually forces aperiodicity

The two Penrose rhombs are usually said to tile the plane only aperiodically. They tile it periodically without difficulty. What cannot be done periodically is tiling them according to the decoration, and the distinction is the whole result.
15 min read 6 figures Order without repetitionGenerated, not drawnSymmetry is decidable

The claim is repeated everywhere: Penrose’s two rhombs tile the plane, and every tiling they make is aperiodic. The first half is true. The second half, stated about the shapes, is false, and the figure below is the counterexample.

The tile does not force aperiodicity — the decoration doesA rhomb with the acute angle of a Penrose tile, repeated by the lattice its own edges generate. The tiling is periodic, so the shape forbids nothing. Adding the edge decoration changes the answer: every interior edge of this tiling presents a double arrow against a single one, which the matching rule refuses.a 72° rhomb, tiled by translation20 tiles, periodic by constructionthe same tiling, decorated31 of 31 interior edges break the rulea shape forces nothing; a decorated shape can72° rhomb
Fig. 1 A rhomb with the acute angle of a Penrose tile, repeated by the lattice its own edges generate. The tiling is periodic — it is a lattice, drawn with a parallelogram cell. On the right, the same tiling with the edge decoration applied: every interior edge presents a double arrow against a single one, which the matching rule refuses.

A rhombus tiles the plane by translation. That is not a subtle fact; it is what a lattice is. So the shapes force nothing, and everything the Penrose result claims comes from the decoration.

What a matching rule is

A matching rule is a marking on the tile edges together with a condition on how marked edges may meet.

For the Penrose rhombs the marking is an arrow on each edge — single or double, pointing one way along the edge — and the rule is that two tiles may share an edge only if they present the same kind of arrow pointing the same way. A tiling that satisfies the rule everywhere is legal; one that does not is not a Penrose tiling, however much it looks like one.

The decoration used on this page is stated explicitly rather than assumed: the two edges meeting at one acute corner carry a double arrow directed away from it, and the two meeting at the opposite acute corner carry a single arrow directed towards it. Conventions differ between sources about which pair gets which marking, and nothing in the argument depends on the choice — what matters is only that the two pairs differ, which is what makes a translated copy fail to match.

That is the mechanism in one line. Translate a rhomb by one of its own edges and the edge it presents to its neighbour is the one on the far side, carrying the other marking. Every interior edge of the translation tiling is a mismatch, and the figure counts them.

What is checked here

The figure’s claim is arithmetic rather than visual, and it runs like this.

Each tile’s four edges are generated with their markings. Edges are collected by their endpoints, so two tiles sharing an edge appear under the same key. Every shared edge is examined and the two markings compared. The assertion is that every shared edge disagrees — not most of them — and the count is printed on the figure.

The assertion is worth having because it would fail under two different errors. If the decoration were applied uniformly, so that all four edges of a tile carried the same marking, every shared edge would agree and the count would come out zero. And if the edge-matching were computed with a floating-point key that missed coincidences, the count of shared edges would drop and the assertion would fire on the total rather than on the disagreement.

What the figure does not check is that the decoration used is the historically correct Penrose one. That is a claim about the literature, stated in the prose above, and the computation is consistent with whatever decoration it is given. Saying so is the honest description: the figure demonstrates that a translation tiling breaks this rule, and the rule’s provenance is a citation.

The thin rhomb, and why the same argument runs

Everything above was said with the thick rhomb, whose acute angle is 72°72°. The thin one, at 36°36°, behaves identically, and running the figure on it is worth a moment because it removes a possible objection.

The tile does not force aperiodicity — the decoration doesA rhomb with the acute angle of a Penrose tile, repeated by the lattice its own edges generate. The tiling is periodic, so the shape forbids nothing. Adding the edge decoration changes the answer: every interior edge of this tiling presents a double arrow against a single one, which the matching rule refuses.a 36° rhomb, tiled by translation20 tiles, periodic by constructionthe same tiling, decorated31 of 31 interior edges break the rulea shape forces nothing; a decorated shape can36° rhomb
Fig. 2 The thin Penrose rhomb, tiled by translation and then decorated. The tiling is periodic for exactly the same reason — a rhombus is a fundamental domain of the lattice its edges generate — and every interior edge again presents a double arrow against a single one.

The objection the second panel forecloses is that the 72°72° rhomb might be special: that a shape with a five-fold-flavoured angle happens to tile periodically while a genuinely Penrose-ish arrangement would not. It is not special. Any rhombus tiles periodically, at any angle whatever, because the argument uses nothing but the parallelogram law. A shape with a 72°72° corner is no more obliged to produce five-fold symmetry than a shape with a 90°90° corner is obliged to produce a square lattice.

That is the general form of the point. A tile’s angles constrain how it can meet its neighbours locally, and constrain nothing about the pattern at large. The whole content of an aperiodicity result is a bridge from the first to the second, and a shape alone does not build one.

Aperiodic sets of tiles, defined properly

The term of art is precise and it is worth having, because the loose version is what produces the error this essay opened with.

A set of tiles is aperiodic when it tiles the plane and no tiling it produces is periodic. Two conditions, and both are required.

The Penrose rhombs with their decoration satisfy both: legal tilings exist, and none of them is periodic. The Penrose rhombs without decoration satisfy the first and fail the second, since the periodic tiling above is one they produce. The pair of shapes is not an aperiodic set; the pair of decorated shapes is.

That distinction is not pedantry, because the interesting mathematical question is exactly the one the decoration answers. Any shape that tiles periodically is uninteresting. The question is whether a local condition — one that can be checked by looking at each junction and nothing more — can force a global property, namely the absence of any translation symmetry however far away. The answer is yes, and it is surprising, and it is the only reason anybody cares.

A Penrose tiling, 5 inflationsTwo rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5
Fig. 3 A legal Penrose tiling, generated by inflation rather than by placing tiles. Every junction here satisfies the matching rule, and no translation maps the pattern onto itself — the two properties an aperiodic tile set has to deliver at once.

Why local rules can force global structure

The mechanism is the same one that makes a Sudoku puzzle have a unique answer, scaled up: a local constraint propagates.

Placing a tile legally against an existing patch leaves fewer choices for the next tile, which leaves fewer for the next. In the Penrose case the propagation is strong enough that a legal patch determines, at every vertex, which of a short list of configurations is present — and those configurations force the beginnings of the next-larger scale of the hierarchy. Applying the argument again forces the scale above that, and the tiling is trapped into the inflation hierarchy whether or not anybody was constructing it that way.

The hierarchy is what rules out periodicity. A periodic tiling has a translation mapping it onto itself; that translation would have to map each level of the hierarchy onto itself; and the levels grow by a factor of φ\varphi each time, so a fixed translation is eventually shorter than the tiles at some level and cannot map that level’s tiles onto other tiles of the same level. The contradiction is at some finite scale, which is why an infinite property follows from a local rule.

InflationOne tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell.one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tilesthe substitution rule applied to a single tile
Fig. 4 The hierarchy itself: one tile becoming several, at successive depths. A legal Penrose tiling is forced into this structure by its matching rules, and a translation symmetry would have to survive at every level of it.

Where the exactness stops

This is the field where the site’s integer machinery does not reach, and the boundary is worth restating on every page in it.

A Penrose tiling has no lattice, so the detector that decides symmetry for the periodic patterns has nothing to enumerate: there is no finite holohedry, no basis in which coordinates are rational, and no exact comparison to make. Everything reported about a Penrose tiling here is a measurement — a tile ratio, a count, a distance — with the measurement stated.

The periodic counterexample is exact and small. The tiling in the figure is a lattice with a parallelogram fundamental domain, and its periodicity needs no proof beyond the definition. That is why it makes a good counterexample: it is the one claim on this page that is certain.

The forcing argument is quoted, not run. That the decorated rhombs admit no periodic tiling is a theorem, proved by the hierarchy argument sketched above, and this site does not verify it. What is verified is a much narrower statement — that one particular periodic tiling breaks the rule at every interior edge — which is a counterexample to the shapes forcing anything, and not a proof of what the decoration forces.

Deflation, which is the argument made precise

The hierarchy sketch above can be turned into a proof, and the step that does it is worth naming because it is the only place the argument becomes rigorous rather than suggestive.

Given a legal tiling, group its tiles into larger copies of the same two shapes. That regrouping — deflation — is the inverse of the inflation that generates the tilings, and the theorem is that a legal tiling admits it uniquely: there is exactly one way to group the tiles into larger ones, and it can be determined locally.

Uniqueness is what makes the periodicity argument work. Suppose a legal tiling had a translation symmetry by some vector t\mathbf{t}. Deflate it: the deflated tiling is legal and still has t\mathbf{t} as a symmetry, since deflation is determined by the tiling and so commutes with any symmetry of it. Deflate again, and again. Each deflation multiplies the tile size by φ\varphi, so after enough steps the tiles are longer than t\mathbf{t} — and a translation shorter than a single tile cannot map a tiling onto itself. Contradiction.

InflationOne tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell.one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tiles4 inflations34 tilesthe substitution rule applied to a single tile
Fig. 5 Five generations of the substitution, each tile becoming several. Deflation is this figure read right to left, and the theorem that a legal tiling can be read that way in exactly one manner is what turns a local rule into a statement about the whole plane.

Two features of that proof are worth extracting. It is a descent argument, of the same shape as the one that forbids five-fold periodicity — assume a symmetry, construct something too small for it to be consistent with. And it needs the hierarchy to be forced rather than merely available, which is the part the matching rules supply and the shapes alone do not.

The generalisation

The question of how few tiles can force aperiodicity has a history of shrinking answers, and it ended recently.

Wang asked in 1961 whether a set of tiles could tile the plane only aperiodically, and conjectured not. Berger disproved him in 1964 with a set of 20 426 tiles, produced by encoding a Turing machine’s halting behaviour into the matching conditions. The count fell steadily — Berger’s own 104, Knuth’s 92, Robinson’s 6 in 1971, Penrose’s 2 in 1974.

Two stood for nearly fifty years, and the remaining question — whether one tile could do it — was answered in 2023. The “hat”, a thirteen-sided polygon found by David Smith, tiles the plane and only aperiodically; a follow-up shape, the “spectre”, does it without needing reflections. Both are undecorated: the shape alone forces it, with no markings at all.

That last point is what makes the einstein result the proper ending to this essay’s story. Penrose’s rhombs need decoration; the hat does not. The matching conditions are built into the shape’s outline, which is exactly what “the tiles force it” was always supposed to mean and had never quite been true of the famous case.

Periodic and aperiodic orderA periodic pattern repeats: there is a translation that maps it exactly onto itself. An aperiodic one does not, and yet it is completely determined and has sharp diffraction — order and repetition are different properties, which is what quasicrystals forced the subject to separate.periodic — a translation maps it to itselfrotation orders limited to 1, 2, 3, 4, 6aperiodic — no translation doesfive-fold symmetry, and sharp diffractionthe restriction assumes periodicity on its first line
Fig. 6 Periodic and aperiodic order side by side. What separates them is not the shapes used but whether any translation maps the whole pattern onto itself — a global property that local rules turn out to be able to control.

The rule is stated edge by edge, and its consequences are easiest to see vertex by vertex.

At a vertex of a Penrose rhomb tiling, the corner angles must sum to 360°360°. The available corners are 36°36° and 144°144° from the thin rhomb and 72°72° and 108°108° from the thick one, so the angle condition alone permits a fair number of combinations — several ways of making 360°360° out of multiples of 36°36°.

The arrow condition cuts that list down sharply, and the surviving configurations are few enough to have been given names: the ace, the sun, the star, the deuce, the jack, the queen, the king. Every vertex of every legal tiling is one of them, and that is the sense in which the rules are local: a patch can be checked for legality by looking at its vertices one at a time, with no reference to anything further away.

The names matter more than they look. Because the list is short, a legal patch can be extended by consulting it — but not always uniquely, and not always successfully. A patch can be legal everywhere and still be impossible to extend, which is the trap anybody tiling by hand falls into: the rules are local, and satisfying them locally is not sufficient to be on the way to a tiling of the plane. Only the hierarchy guarantees that, and the hierarchy is not something a person laying tiles can see.

The surprising part

Undecidability is lurking underneath all of this, and it is the reason Wang’s conjecture was worth asking about.

Wang’s question was motivated by a decision procedure. He observed that if every tile set that tiles the plane also tiles it periodically, then there is an algorithm to decide whether a given set tiles at all: search for a periodic tiling, and search for a proof that no tiling exists, and one of the two searches must terminate. So “no aperiodic tile sets” would have given a decision procedure for the domino problem.

Berger proved the converse. Aperiodic sets exist, the domino problem is undecidable, and the two results are the same result — his construction encodes a Turing machine into a tile set in such a way that the tilings correspond to computations.

Which puts a Penrose tiling in unexpected company. It is a picture of a computation that never halts, laid out in the plane, and the reason it never repeats is the reason a non-halting computation never repeats its state. That is the connection worth carrying out of this essay: aperiodicity and undecidability are the same phenomenon in different clothes, and the golden ratio is a detail of one especially pretty instance.

Who found what

Wang posed the question in 1961; Berger, his student, answered it in 1964. Robinson’s 1971 six-tile set is the one usually shown, because its forcing argument is short enough to follow.

Penrose found his tilings in 1974, working from the pentagon rather than from the decidability question, and the rhomb version with arrows came shortly after. De Bruijn supplied the algebraic explanation in 1981 with the cut-and-project construction, which turned the tilings from a curiosity into an instance of something general — a year before Shechtman measured a material that needed it.

Smith’s hat was found in November 2022 by a retired print technician working with a shape-exploring program, and was proved aperiodic in 2023 with Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss. Goodman-Strauss is a co-author of the book that gave orbifold notation its modern form, which is a small illustration of how few people work on this.

Where the ladder goes next

The tilings themselves are Penrose tilings, and the construction that generates them here is inflation.

The property they are the standard example of is order without periodicity, and the algebraic explanation of where they come from is cut and project — which starts from the higher-dimensional lattice where five-fold symmetry is permitted.

The contrast case, where a pattern’s symmetry is decided rather than measured, is the orbit and its detector.

What the pictures here cannot show. The right-hand panel marks every interior edge of a finite patch as a rule violation. That no infinite legal tiling is periodic is a theorem about all tilings, and no picture of one tiling — legal or otherwise — can establish it. The counterexample on this page is a counterexample to a different and weaker claim, and keeping the two apart is the point of the essay.

Where this goes next

890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5 Penrose tilings Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice. one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tilesthe substitution rule applied to a single tile Inflation, and where the golden ratio comes from A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio. 181 reflections10-fold symmetryforbidden to any latticeinteger combinations of ten star vectorsaperiodic Order is not periodicity For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.