Crystallography & Symmetry
patterns derived from their groups
Concept

Vertex species — where it appears

The cyclic sequence of polygon sizes meeting at a vertex of a tiling, counted up to rotation and reflection. Twenty-one of them satisfy the angle equation and ten tile nothing at all, refuted by counting places round a polygon rather than by measuring.

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

Every way regular polygons can fill a turn. The seventeen multisets of regular polygons whose interior angles add to exactly 360°, listed with the sum that qualifies each of them. They are found by a search over sizes from three upward: the largest polygon that can appear is the forty-two-gon, which needs a triangle and a heptagon beside it, and the search stops there because the smallest interior angle is a third of a turn so at most six polygons can meet. Nothing here is a table looked up — the list is the output of the search, and every count on the page downstream of it is counted from this one.

Twenty-one vertices, eleven tilings

Regular polygons meeting at a point must fill exactly a turn, which is a Diophantine equation with seventeen answers and twenty-one cyclic arrangements. Ten of the twenty-one tile nothing at all — and the argument that kills them counts places round a polygon rather than measuring anything.

classification · Tilings 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

Named alongside it

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

Archimedean tilingWallpaper groupAccidental symmetryCase analysisDecidabilityDetectionEdge to edgeEnumerationOrbitParityProof by contradictionQuadratic field

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