Crystallography & Symmetry
patterns derived from their groups
Concept

Refutation — where it appears

An argument that establishes no object of a given kind exists, as opposed to a search that has not found one. It settles a question in the direction of impossibility at one stroke and costs nothing per candidate, which is why a count closed by a refutation and a construction is closed and one closed by construction alone is a lower bound.

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

a general quadrilateral tiles. A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested.

Which shapes tile by themselves

Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.

classification · Tilings The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together.

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

classification · Decidability

Named alongside it

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

Plane groupCensusConvexityCountingEnumerationThe Euler characteristicExhaustive searchHalf-turnMonohedral tilingOrbitParallelogonParity

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