Two tiles that never repeat

Take a floor and cover it with copies of a single shape — square tiles, or hexagons — and the pattern repeats: slide the whole thing over by one tile and it lands back on itself. That sliding-repeat is called being periodic, and almost every tiling you have ever seen is periodic.

In 1974 the mathematician (and later Nobel laureate) Roger Penrose found something astonishing: a way to tile the whole infinite plane using just two tile shapes, with no gaps and no overlaps, such that the pattern never repeats — there is no slide at all that maps the tiling back onto itself. These are the Penrose tilings, and they are the most famous aperiodic tilings in mathematics.

The secret is not the tiles alone but the matching rules — markings on the edges that say which edge may touch which. With those rules the two tiles can still cover the plane, but only non-periodically. Hidden inside every Penrose tiling you find five-fold symmetry (the symmetry of a five-pointed star) and the golden ratio \varphi — the very fingerprints a repeating tiling is forbidden to have.

The two shapes: kite & dart, or two rhombs

Penrose tilings come in two standard tile sets, and both are carved straight out of the regular pentagon, whose corner angle is 108^\circ:

• the kite and dart — two four-sided pieces you get by cutting up a pentagon; and
• the two rhombs — a thick rhombus and a thin rhombus. This is the set we will focus on, because its angles are the easiest to read off.

A rhombus is a "pushed-over square": all four sides equal, opposite angles equal. So each rhomb is described by just one number — its sharp (acute) corner angle — and the other corner is whatever is left to make a straight 180^\circ along the side.

Worked example 1 — the angles of the two rhombs

Every angle in a Penrose rhomb is a multiple of 36^\circ — one fifth of a half-turn — because 36^\circ = 180^\circ / 5 is the pentagon's natural unit.

The thick rhombus. Its sharp corner is 72^\circ (that is 2 \times 36^\circ). The other corner completes the straight line beside it:

So the thick rhombus has angles 72^\circ,\,108^\circ,\,72^\circ,\,108^\circ — and 108^\circ is exactly the interior angle of the regular pentagon.

The thin rhombus. Its sharp corner is 36^\circ (a single 36^\circ), so its blunt corner is

The thin rhombus therefore has angles 36^\circ,\,144^\circ,\,36^\circ,\,144^\circ. Notice every one of the eight angles — 36, 72, 108, 144 — is a whole number of 36^\circ steps. That is what lets so many of them fit snugly around a point.

Worked example 2 — the tile ratio is the golden ratio (and why that forbids repeating)

Count the tiles in a large Penrose patch. As the patch grows, the ratio of thick rhombs to thin rhombs settles down to a single fixed number — the golden ratio:

(The kite-and-dart set does the same: the ratio of kites to darts also tends to \varphi.) Now here is the punchline. The golden ratio is irrational — it can never be written as one whole number divided by another.

Suppose, for a moment, the tiling were periodic. Then it would have a repeating unit cell — a fixed block that copies across the plane — and that block would contain some whole number of thick rhombs, say a, and some whole number of thin rhombs, say b. The overall ratio would then have to be the fraction

But we just said the ratio is \varphi, which is irrational. A number cannot be both a whole-number fraction and irrational, so the assumption breaks. There is no unit cell — the tiling cannot be periodic. The golden ratio quietly proves the aperiodicity all by itself.

See the two rhombs and a five-fold star

Step through the figure. First meet the thick rhombus with its 72^\circ / 108^\circ angles, then the thin rhombus with its 36^\circ / 144^\circ angles. The coloured corner dots stand for the matching rules — only like-coloured corners are allowed to meet. Finally, watch five thick rhombs slot around a single point: 5 \times 72^\circ = 360^\circ closes the circle exactly, and a five-fold star — forbidden to any repeating tiling — springs into being.

Why five-fold symmetry forces aperiodicity

A repeating (periodic) tiling can only have rotational symmetry of order 2, 3, 4 or 6 — that is the crystallographic restriction. Five-fold symmetry is precisely the one it forbids: you simply cannot make a repeating pattern that looks the same after a 72^\circ turn.

But Penrose tilings are full of local five-fold (and ten-fold) symmetry — the star you just built is only the simplest example. A pattern that shows a symmetry no repeating pattern is allowed to have must not be repeating. So five-fold symmetry is a second, independent proof that Penrose tilings are aperiodic — arriving at the same verdict as the golden-ratio argument.

A little history, and where this leads

Penrose tilings began as a beautiful mathematical curiosity, but in 1982 Dan Shechtman discovered real crystals — quasicrystals — whose atoms are arranged in exactly this kind of ordered-but-non-repeating way, with the "impossible" five-fold symmetry. The discovery was so contrary to textbook crystallography that it took years to be accepted; it won Shechtman the Nobel Prize in Chemistry in 2011.

The ideas here open onto a whole subject: how a Penrose tiling can be grown by substitution and inflation (repeatedly subdividing each tile), how it can be produced as a slice of a higher-dimensional lattice by the cut-and-project method, the modern hunt for aperiodic monotiles (a single tile that tiles only aperiodically), and the physics of quasicrystals.