Crystallography & Symmetry
patterns derived from their groups
Concept

Aperiodic tile set — where it appears

A set of shapes that tiles the plane and admits no tiling with a period. The first had over twenty thousand tiles, the smallest set of Wang tiles has eleven, and the smallest number of shapes is one — which is what makes such a set a statement about geometry rather than about counting.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The hat: eight kites, thirteen sides. The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search.

One tile, and no period

Every aperiodic pattern in this collection so far needs two shapes. A search over the eight-hundred-and-seventy-three ways of gluing eight kites together, filtered by nothing but whether a shape tiles and whether it repeats, returns exactly one — and it is the shape announced in 2023.

aperiodic · Monotile Unreflected copies stop at 1 ring. Copies of the hat, all of the same handedness, covering a core of 1 ring of hexagons — 9 tiles, every cell covered once. At 2 rings the same search runs to exhaustion and returns nothing: there is no such covering, and the failure is a proof for that region rather than a search that gave up. The reflected copy is not a convenience of the drawing; the tiling cannot proceed without it.

The tile that needs no reflection

One shape tiles the plane and never repeats, and it does it with copies of both hands. Cut the tiles out of card and that is nothing; ask for it in a molecule, where handedness cannot be undone by turning something over, and it is the whole question.

aperiodic · Monotile 656 sets, every one decided. Every set of one, two, three and four tiles over two colours — sixteen tiles exist in all, so these are complete lists rather than samples — reduced by relabelling the two colour alphabets, and each set decided by the two half-searches. The last column is the one that matters: it is empty. At these sizes there is no room for a set that tiles the plane and admits no periodic tiling, which is the residue undecidability lives in. The smallest aperiodic set is known to have eleven tiles and four colours.

How much room a hard question needs

No algorithm decides whether a set of tiles covers the plane. Every set of four or fewer tiles over two colours is nevertheless decided here, exhaustively, in under a second — because the sets that defeat the two half-searches have nowhere small to live.

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

Named alongside it

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

AperiodicityMonotilePolykiteDecidabilityLaves tilingMatching rulesPatch frequencyChiralityDecorated tileEdge to edgeEnantiomorphExhaustive search

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