Crystallography & Symmetry
patterns derived from their groups
Concept

Matching rules — where it appears

Constraints on how tiles may meet, decorating their edges so that only some arrangements are legal. The Penrose rhombs tile periodically without difficulty; what cannot be done periodically is tiling them according to the decoration.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

A Penrose tiling, 4 inflations. Two 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.

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.

aperiodic · Aperiodic The tile does not force aperiodicity — the decoration does. A 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.

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.

aperiodic · Aperiodic 8 tiles over 5 colours. Wang tiles: unit squares with a colour on each edge, which may be laid side by side only where the touching edges agree, and which may never be turned or reflected. That last restriction is what makes them a computational object rather than a jigsaw — an edge colour is a symbol passed from one tile to its neighbour, and turning a tile would let a symbol change direction. The set here was generated from a stated seed.

Nothing decides whether a set of tiles tiles the plane

This collection rests on decidability — generate a pattern, forget the group, rediscover it, compare. One question in the same subject has no procedure at all: given a finite set of tiles, whether they cover the plane cannot be decided by any algorithm whatever. What can be done is two half-searches, and measuring what they leave behind.

classification · Decidability One hat patch laid out as hats, as equilateral tiles and as turtles. A patch of 36 hats found by exact cover on the kite grid, 4 of them reflected and drawn in the second colour, laid out three times. Every edge keeps its direction; short edges and long edges are given their own lengths. At short 1 and long √3 the tiles are hats, at equal lengths they are the equilateral member of the family, and at short √3 and long 1 they are turtles. In all three the same tiles touch the same neighbours along the same edges, and each layout was checked to be a tiling: 0 gaps and 0 overlaps, 0 gaps and 0 overlaps, 0 gaps and 0 overlaps among 1500 sample points, and every interior vertex surrounded by a full turn.

The hat and the turtle are one tiling

The hat has short sides and long sides; the turtle has the same turns with the two lengths exchanged, and looks nothing like it. Take a patch of hats, keep every edge pointing the way it points, stretch the short edges and shrink the long ones, and the patch becomes a patch of turtles — every tile touching the same neighbours along the same edges.

aperiodic · Monotile Every vertex of a Penrose patch, by the symmetry of what lies within three edges of it. A region of a Penrose rhomb tiling 15.2 edges across. Each vertex is marked by the local group of the patch of radius 3.05 edges about it: a short line along the mirror where that patch has one (98 vertices), a large dot where it has a five-fold centre (7), and a small dot where it has no symmetry at all (118). The mirrors are common and point in the tiling's ten directions; the five-fold centres are rare; nothing has a half-turn.

Five-fold centres stay few, and mirrors do not

The Fibonacci chain has one palindrome of every even length and two of every odd length, for ever. Run the same census on the patches of a Penrose tiling and the plane gives three different answers at once. Patches with a five-fold centre stop at four, however large they are drawn. Patches with a mirror grow in number with their radius, by about fifty for every edge. And no patch wider than a single tile has a half-turn at all. What decides which answer an operation gets is the size of the set it leaves fixed in internal space.

aperiodic · Complexity The error in the count, against tiles chosen at random. For clusters from about twenty-five hats to nearly two trillion, the distance of the reflected count from the exact share of all hats. For the hat it stays under one half. For independent random choice at the same share the expected error grows as the square root of the size, past 400,000 at the largest cluster.

The count of reflected hats is never one out

In a hat tiling one tile in about eight is reflected, and that ratio was only ever quoted here. Four clusters and a table of small integers give it exactly: seven plus three root five over two, the fourth power of the golden ratio. The same table says something the ratio does not. Grow a cluster to two trillion hats and its reflected count is still within half a tile of the exact share, where the same share chosen at random would be out by nearly half a million.

aperiodic · Monotile

Named alongside it

The objects these essays reach for when they reach for this one.

AperiodicityAperiodic tile setLocal rulesMonotilePenrose tilingsAverage structureBragg peakCertificateCut-and-projectDecidabilityDecorated tileEdge to edge

All concepts