Tiles that never repeat

A floor tiled with squares repeats: shift the pattern one tile to the right and it lies exactly on itself. So does a floor of hexagons, or of any of the thousands of shapes that tile the plane in the obvious ways. For a long time it was believed that repetition was unavoidable — that any set of shapes able to tile the whole plane could also tile it periodically. Hao Wang conjectured as much in 1961, and Robert Berger disproved it in 1966 with a set of 20,426 square tiles with coloured edges that tile the plane only non-periodically. The number came down over the next decade to a few dozen, then to six. In 1974 Roger Penrose found two.

Penrose’s tiles come in several versions. The one drawn here uses two rhombs with angles from the regular pentagon: a thick rhomb with angles $72°$ and $108°$, and a thin one with angles $36°$ and $144°$. They tile the plane with five-fold stars and decagonal rings everywhere, and never with a period. This essay builds the tiling by the simplest route there is — cutting shapes into smaller copies of themselves — and then shows where the non-repetition comes from. It comes from a number, and the number is $\varphi$.

Two rhombs, four halves #

Cut each rhomb in two across one diagonal. The thin rhomb, cut across its short diagonal, gives two triangles with angles $36°$, $72°$, $72°$ — the golden triangle, the shape of a pentagram’s point. The thick rhomb, cut across its long diagonal, gives two triangles with angles $36°$, $36°$, $108°$ — the golden gnomon. Those are exactly the two triangles that the golden triangle is cut into when a base angle is bisected, and that is not a coincidence: the whole construction runs on that cut.

The rule, due to Raphael Robinson, cuts each half into smaller halves of both kinds. A thin half splits into a thin half and a thick half; a thick half splits into two thick halves and a thin one. Every new piece is smaller than its parent by a factor of $\varphi$ in length, so every piece in every generation is one of the same two triangles at a smaller scale. Repeating the cut forever refines a region into ever smaller tiles; repeating it and then scaling up by $\varphi$ each time grows a tiling that covers ever more of the plane with tiles of fixed size.

Growing the patch #

Start from ten thin halves arranged in a star around a point, and cut.

After three cuts there are already recognisable rhombs, stars and a decagonal ring round the centre. After five cuts, in the first figure, the patch is a disc of 890 half-rhombs with five-fold rosettes scattered throughout. The pattern looks as though it ought to repeat — it is full of the same local arrangements — and it does not. No two positions in it have the same surroundings out to arbitrary distance, and as the essay’s argument will show, no tiling built this way can repeat at all.

What the construction guarantees without further argument is that it tiles: every cut replaces a triangle by triangles that fill it exactly, so after any number of cuts the pieces fill the original star exactly, and scaled up they fill ever larger discs. Taking a limit — carefully, by a compactness argument that picks a nested sequence of patches — gives a tiling of the entire plane.

Counting the tiles #

The rule says how many pieces of each kind each piece becomes, so the counts can be followed without drawing anything.

In matrix form the rule is: the pair (thin, thick) is multiplied by the matrix with rows $(1, 1)$ and $(1, 2)$ at every cut. Both columns of the table are every other Fibonacci number times ten — $10, 10, 20, 50, 130, 340, 890$ thin halves and $0, 10, 30, 80, 210, 550, 1440$ thick — and the ratio thick ÷ thin runs $1.0, 1.5, 1.6, 1.6154, 1.6176, 1.6180$, climbing towards $\varphi$.

The reason is the direction a matrix leaves alone. The matrix’s larger eigenvalue is $\varphi^2 = 2.618$, and its eigenvector is in the ratio $1 : \varphi$. Applying the matrix over and over swings any starting vector onto that direction, so whatever mixture of halves the tiling starts from, the proportions converge to one thin half for every $\varphi$ thick halves. And since each rhomb is two halves of the same kind, thick rhombs outnumber thin rhombs by the golden ratio, in any large enough region of any tiling built by this rule.

Why an irrational ratio forbids a period #

That ratio is the whole of the aperiodicity argument, and it is short. Suppose some tiling by these rhombs were periodic: suppose some shift carried it onto itself, and so did a second shift in a different direction. Then the tiling would be a repeating pattern, the same parallelogram-shaped block of tiles copied across the plane, and in any large region the proportion of thick tiles to thin would be the proportion inside one block — a ratio of two whole numbers. A periodic tiling has a rational ratio of thick to thin tiles.

The tilings built by the rule have ratio $\varphi$, and $\varphi$ is not a fraction. So none of them is periodic. The figure measures the ratio directly, in discs of growing size in one large patch: it wanders at first, when a disc holds only a few dozen tiles and one more of either kind changes it noticeably, and then settles towards $\varphi$ as the disc grows. The pentagon’s incommensurable diagonal, the golden triangle’s cut, and the tiling’s refusal to repeat are one fact seen three times.

The argument needs one more ingredient to be complete, because a set of tiles might tile the plane in several different ways, and the ratio argument only covers the tilings the rule builds. What makes it cover every tiling is that the process runs backwards uniquely: in any tiling that obeys Penrose’s rules, the tiles can be grouped into larger tiles of the same two shapes in exactly one way. So every such tiling is, at every scale, a tiling built by the rule, and every one has thick-to-thin ratio $\varphi$.

There is a second way to see the irrationality, found by Nicolaas de Bruijn in 1981. Every Penrose rhombus tiling is a shadow of an ordinary periodic lattice in five dimensions: take the five-dimensional grid of whole-number points, cut a thin slab through it at an angle whose slope involves $\varphi$, and project the lattice points inside the slab onto a plane. The projected points are the vertices of a Penrose tiling. The lattice upstairs is as periodic as anything can be; the tiling downstairs inherits no period, because the slab’s slope is irrational and so its intersection with the lattice never repeats — exactly as a line of irrational slope through a grid never repeats its word.

The rules that make the rhombs behave #

There is an honest complication, and it is worth stating because it is often left out. The two rhombs, as bare shapes, can tile the plane periodically: the thick rhomb alone tiles the plane in parallel strips, like any parallelogram. What prevents that is not the shapes but a matching rule — a decoration on the edges, such as arrows or arcs, that must line up where two tiles meet. Penrose’s rules forbid every arrangement except those that decompose uniquely into larger tiles, and so, by the argument above, forbid every periodic one.

The triangles in the construction carry the rules implicitly: the subdivision always puts the pieces together in the same orientations, and a tiling grown from it automatically obeys the decorations. That is why the construction gives only non-periodic tilings without the rules ever being drawn. But a tiler handed a box of plain rhombs could make a periodic floor, and the claim “Penrose’s two tiles tile only non-periodically” is always a claim about the decorated tiles.

For half a century the natural question was whether a single shape could do what Penrose’s pair does, with no decorations at all — an “einstein”, German for one stone. In 2023 David Smith, Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss found one: a thirteen-sided polygon nicknamed the hat, which tiles the plane, and only non-periodically, by its shape alone. Their proof uses the same ingredient as Penrose’s: the hat’s tilings group uniquely into larger clusters that behave like larger tiles, and the substitution matrix has an irrational eigenvalue ratio.

A hierarchy at every scale #

Six cuts show the structure the counting hides. The tiling is a hierarchy: groups of tiles form larger versions of the same two shapes, groups of those form larger ones again, and the pattern at each scale is the pattern at the scale below, magnified by $\varphi$. The small five-pointed stars made of five thick rhombs sit inside larger stars made of whole clusters, and those inside larger ones. That hierarchy is exactly the substitution run backwards, and it is what the uniqueness of composition says: there is only one way to read any Penrose tiling as a tiling by larger tiles, and so the whole hierarchy is determined by the tiles themselves.

It is also why the pattern cannot repeat, seen from another side. A period is a fixed shift that carries the tiling onto itself. Carry that shift up the hierarchy and it becomes a shift of the larger tiles — but at a large enough scale the tiles are bigger than the shift, and a shift smaller than a tile cannot carry a tiling onto itself. So a period would have to be zero.

Kites and darts, the same tiling in other clothes #

Penrose’s best-known version uses different tiles, a kite and a dart, cut from a rhombus with angles $72°$ and $108°$ along a line that divides its long diagonal in the golden ratio. Both are made of the same two triangles as the rhombs, the golden triangle and the golden gnomon, in different arrangements, and the two tilings convert into each other by cutting and regluing tiles locally. In a kite-and-dart tiling, kites outnumber darts by $\varphi$, for the same eigenvector reason.

Nothing about the argument depends on which pair is used, and that is the point: the aperiodicity lives in the substitution and its matrix, not in the particular shapes. Any pair of tiles whose substitution matrix has an irrational ratio of eigenvector entries, and whose matching rules force the substitution, tiles only non-periodically. The golden ratio appears because the pentagon’s triangles have the simplest such substitution there is, with the matrix of the Fibonacci numbers.

Order that diffracts #

A crystal is recognised by its diffraction pattern: shine X-rays through it, and the lattice’s periodic rows scatter them into sharp spots, which is a Fourier transform of the arrangement of atoms. Sharp spots were long taken as proof of periodicity, and periodicity forbids five-fold symmetry, as the trace of a rotation by 72° shows.

In 1982 Alan Mackay placed small dots at the vertices of a Penrose tiling and shone laser light through a photographic slide of the pattern. The result was a pattern of sharp spots with ten-fold symmetry — sharp because the tiling is perfectly ordered, ten-fold because its local symmetry is five-fold, and impossible for any lattice. In the same year, and independently, Dan Shechtman saw the same kind of pattern from an aluminium–manganese alloy, and the quasicrystals that were then found are understood as three-dimensional relatives of Penrose’s tiling, built from rhombohedra with golden-ratio proportions and carrying the icosahedral symmetry of the regular solids that need $\sqrt 5$.

Everything that happens, happens everywhere #

Non-periodic does not mean disordered. Every Penrose tiling has a strong form of order called repetitivity: any finite patch that occurs anywhere in the tiling occurs again within a bounded distance of every point. John Conway made it quantitative — a patch of diameter $d$ appears again within a distance of roughly twice $d$ from anywhere — so a walker who has seen a patch will meet it again soon, wherever they go.

There are uncountably many different Penrose tilings, no two of them related by a shift or rotation, and yet every finite patch of one occurs in all of the others. So no finite observation can tell two Penrose tilings apart. That combination — perfect local agreement with global difference — is the geometric shadow of the symbolic fact that every Sturmian word of one slope contains the same blocks with the same frequencies. The connection is closer than an analogy. Decorate the rhombs with the short line segments Robert Ammann found, and in any Penrose tiling the segments join up into five families of parallel straight lines; within each family the gaps between neighbouring lines are of two lengths, long and short in the ratio $\varphi$, and they occur in exactly the order of the letters of the Fibonacci word.

What the patches cannot show #

They cannot show the whole plane. Every figure is a finite patch grown from a star, and the statements about all tilings — that each one’s ratio is $\varphi$, that none is periodic, that every patch recurs — are theorems about infinite objects. The figures show the construction, the counts that converge and the ratio settling in growing discs; the passage to infinity is argued, not drawn.

They cannot show the matching rules. The rhombs are drawn undecorated, and the construction obeys the rules because it is built from the rule; a figure of undecorated rhombs cannot show why plain rhombs fail and decorated ones succeed.

And they cannot show uniqueness of composition. That every tiling obeying the rules groups uniquely into larger tiles is the step that turns “the tilings the construction builds are non-periodic” into “every tiling is non-periodic”, and it is a case analysis of how tiles can surround a vertex, which no single picture contains.

Where tilings lead: a question no algorithm answers #

Wang conjectured that every set of tiles that can tile the plane can tile it periodically, and his reason is the best way to understand what Berger’s counterexample means. If Wang had been right, there would be an algorithm to decide whether a set of tiles tiles the plane: search for larger and larger square patches that can be tiled, and in parallel search for a periodic tiling. If the tiles do not tile, the first search eventually fails at some size; if they do, the second eventually finds a period. One of the two searches must succeed.

Berger proved in 1966 that no algorithm decides whether a set of tiles tiles the plane — the tiling problem is undecidable, by an encoding of any computer program into a set of tiles that tiles exactly when the program runs forever. So Wang’s search cannot always finish, which means there must be sets of tiles that tile the plane and never periodically. Aperiodic tile sets are not curiosities that happen to exist; they are forced to exist by the undecidability of tiling, and Penrose’s two tiles are the simplest witnesses. That the most beautiful example is governed by the golden ratio is a gift of the pentagon.

Two shapes, one irrational number #

Cut a thin golden half-rhomb into one thin and one thick half, and a thick half into two thick and one thin, each smaller by $\varphi$, and repeat: the pieces fill the plane as Penrose’s rhombus tiling. The counts of each kind follow the matrix with rows $(1, 1)$ and $(1, 2)$, so they are every other Fibonacci number, and the ratio of thick to thin tiles tends to its eigenvector’s ratio, $\varphi$.

A periodic tiling would have a rational ratio, so no such tiling repeats; matching rules on the edges and the uniqueness of grouping tiles into larger ones extend that to every tiling the decorated rhombs allow. The tilings are nonetheless highly ordered, with every patch recurring near every point. And aperiodic tiles must exist at all because deciding whether tiles tile the plane is beyond any algorithm.

A pattern whose proportions are irrational cannot repeat — and a rule that forces an irrational proportion forces a pattern that never does.