Crystallography & Symmetry
patterns derived from their groups
Concept

Uniform tiling — where it appears

A tiling whose symmetry group carries any vertex onto any other, which is stronger than every vertex merely looking alike. The two of the eleven admitting non-periodic variants are exactly the ones where that distinction does work rather than merely passing.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The cell of 3.4.6.4, and the vertices in it. 3.4.6.4 drawn with the cell its own translations define. The lattice is hexagonal and the cell holds 6 vertexes, marked. Neither was chosen: the translations are the vertex-to-vertex vectors that carry every polygon of the patch onto a polygon of the patch, and the cell is the shortest independent pair of them. Expressed in that basis the vertices have coordinates that are exact and are not fractions — a vertex of this tiling sits at 1/(1 + √3) of a cell — which is why the detector that decides its group works in ℚ(√3) rather than in the rationals.

Eleven tilings, five groups

Hand each of the eleven uniform tilings to a detector that has never heard of tilings and ask what its symmetry is. Six of them answer p6m. Twelve of the seventeen wallpaper groups never appear at all — and the coordinates the question has to be asked in are not fractions.

classification · Tilings 3.4.6.4 and its dual. The tiling in pale outline with its dual drawn over it: one dual vertex at the centre of every tile, one dual edge across every shared edge, and one dual tile round every vertex. 3.4.6.4 has 3 kinds of tile and one kind of vertex; its dual has one kind of tile and 3 kinds of vertex, and the congruence of those tiles is checked rather than eyeballed — every dual face presents the same cyclic sequence of squared edge lengths, compared exactly. That swap is what the eleven duals are for: read one way the list classifies tilings with all vertices alike, read the other it classifies tilings with all tiles alike.

Eleven duals, one tile each

Swap the vertices of a uniform tiling for its tiles and the eleven come back as eleven tilings by a single repeated shape. Three of those shapes are pentagons — which is worth pausing over on a site whose other essays prove that five-fold symmetry cannot exist.

classification · Tilings The square net drawn as the Cayley graph of four different groups. The same square net four times. In each panel the edges are coloured by the group operation that spells them, read from one regular subgroup of the net's symmetry group: the lattice of translations, a p2 whose half-turn centres sit at the midpoints of edges, a pmm whose four mirrors run between rows of vertices, and a p4 whose quarter-turn centres sit in the middle of the squares. No symmetry element drawn passes through a vertex, which is what makes each group act without fixing any, and the large dot is the vertex named by the identity.

The square net draws eleven groups

The Cayley graph of a wallpaper group is a periodic net. Asked the other way, the question is which nets are Cayley graphs, and of what. A net is one exactly when some group of its symmetries carries every vertex to every other and fixes none. All eleven uniform nets pass. The square net is the Cayley graph of eleven of the seventeen groups, the kagome net of exactly one, and between them the eleven nets use all seventeen.

operations · Presentations

Named alongside it

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

Archimedean tilingStabiliserWallpaper groupAccidental symmetryCayley graphCrystal netCrystallographic restrictionDetectionDualityEdge to edgeFree actionGenerating set

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