Crystallography & Symmetry
patterns derived from their groups
Concept

Rotoinversion — where it appears

A rotation followed by an inversion through a point on its axis, a single operation although neither half of it need be a symmetry. Its linear part fixes no direction at all, which is why a class containing one can never act on space without holding a point still.

Named by 4 essays across 4 fields — each of them below, with the objects they name alongside it.

The crystal classes 4, 4̅, 3̅, 6̅. 4, 4̅, 3̅, 6̅: the orbit of a general direction under each group, giving 4, 4, 6, 6 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

What a trace decides

Ten kinds of operation, and two integers tell them apart. The determinant and the trace name a symmetry operation completely — which is why the classification can be run on integer matrices in a lattice basis and never once ask what angle anything turns through.

point-groups · Crystal classes P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn.

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

operations · What symmetry is P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra.

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

space-groups · Space groups Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups

Named alongside it

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

Screw axisSymmetry operationCartan dieudonneCentringClosureCompositionCrystal classCrystallographic restrictionDecidabilityDeterminantFinite groupFixed point

All concepts