Crystallography & Symmetry
patterns derived from their groups
Concept

Conway criterion — where it appears

A condition on a tile's boundary that guarantees it tiles the plane. The closed path must cut into six arcs, one a translate of another and four carried onto themselves by half turns. It is sufficient and not necessary, and it names the group that builds the tiling rather than asserting one exists.

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

Also named here as isohedral — the same set of essays touches all of them, so they are one junction rather than several.

Y-pentomino: A B C D E F, with 6 arcs. The boundary of the Y-pentomino cut into six arcs. A runs from one corner to another and D is the same arc traversed backwards, so D is a translate of A and the translation is (3, 1) cells. Each of B, C, E and F is carried onto itself by the half turn about its own midpoint, and those midpoints are the four marked dots — That is Conway's criterion, and a shape meeting it tiles the plane by translations and half turns.

A tiling of the whole plane, decided on one tile's edge

Whether a shape tiles the plane is a question about an infinite object, and there is no procedure that answers it. There is a procedure that answers it *sometimes*, and it reads nothing but the shape's own boundary — a closed path of a few dozen steps, cut into six arcs. When the cut exists the tiling exists, and the cut names the group that makes it.

classification · Isohedral anisohedral: 2 orbits of congruent tiles. A tiling of the plane by 8 copies of one shape per cell of a lattice of index 64, drawn 1 cell across and 8 up, and coloured by which orbit of the tiling's own symmetry group each tile belongs to. The group has 4 operations per cell and 2 orbits: every tile is congruent to every other, and no motion of the whole pattern carries a tile of one colour to a tile of another. Congruence is a fact about the shapes; an orbit is a fact about the pattern, and they are different facts.

One shape, two kinds of tile

A tiling by copies of a single shape looks as though it must be homogeneous — every tile is congruent to every other, so what could distinguish them? The symmetry group can. There are shapes that tile the plane and admit no tiling whose group carries any tile to any other, and the smallest of them has eight cells.

classification · Isohedral

Named alongside it

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

IsohedralMonohedral tilingTiling by a groupAnisohedralClassificationDecidabilityForcingHalf-turnLocal rulesOrbitStabiliser

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