Crystallography & Symmetry
patterns derived from their groups
Concept

Arithmetic class — where it appears

A point group together with the lattice type it acts on, taken up to a change of integer basis. There are seventy-three in space and thirteen in the plane. It is the unit the space groups are built from, because point operations and translations have to be compatible as integer matrices rather than merely as shapes.

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

21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

aperiodic · Modulation Seventy-three arithmetic classes, from fourteen groups. Every subgroup of every lattice's own group, split by whether the subgroup's own Bravais group is that lattice's. The ones that are not belong to a lower lattice and are counted there, which is what stops the same class being counted twice. The running total ends at seventy-three, and no conjugacy in GL(3, ℤ) was ever decided.

Seventy-three, without a search

The unit a space group is built from is a point group together with the lattice it acts on, and there are seventy-three of them. Getting there looks like it needs conjugacy in GL(3,ℤ), which is a search this collection tried and abandoned. It does not: every finite group of integer matrices carries a canonical larger group that says which lattice it belongs to, and once that is computed the search has nothing left to do.

point-groups · The fourteen Bravais lattices

Named alongside it

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

Bravais latticeCohomologyCrystal classGram matrixGroup extensionHolohedryModulationNormaliserOrigin shiftPlane groupPoint groupReciprocal lattice

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