Crystallography & Symmetry
patterns derived from their groups
Concept

Improper operation — where it appears

An operation that reverses handedness — a mirror, the inversion or a rotoinversion — so that its matrix has determinant minus one. No solid object can perform one on itself, and every group containing one is either its rotations with the inversion added or a rotation group with half its operations negated.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

A triangle sliding from one mirror to another. Two vertices fixed and the third slid along a line. At the left end the triangle is isoceles about the vertical, at the right end it is isoceles about a different line, and both ends have a mirror — so the measure is exactly zero at both, by cancellation rather than by a search running out. Between them the triangle has no mirror at all and the measure rises to 0.0422. The faint curve is the handedness over the same family, which is a different quantity: it has a sign, it is largest where the measure is not, and it does not vanish at either end.

How chiral, as a number

A group answers one bit: a set either has an improper symmetry or it does not. Two shapes can both be chiral and one of them be a mirror-symmetric thing with a substituent out of place while the other is a helix, and nothing in the classification says which is which. A distance does.

space-groups · Chirality 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 Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 80 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet.

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

space-groups · Chirality The seven edges a sheet can have. The seven frieze groups drawn as the profile of a cut edge: along the strip is along the edge, up the strip is the sheet's normal, and the half-sheet itself stretches away from the viewer. Each is labelled with what its operations are in the sheet — a mirror standing across the edge, a mirror or glide in the sheet's own plane, a half-turn about the cut's in-plane normal — and with how many of the eighty layers have it as the edge in some direction. Only p1 and p2 have no mirror image among their operations, so only they are chiral edges; every one of the seven occurs.

Cutting a sheet can give it a hand

Cut a periodic sheet along a line and the edge that is left has a symmetry of its own: a frieze group, drawn in the plane across the cut. A chiral sheet can only have chiral edges. An achiral one usually has chiral edges too, in all but a few directions — and eight of the eighty layer groups have no achiral edge at all, however the sheet is cut.

space-groups · Chirality Two cuts that give back the mirror a single cut takes away. A sheet whose only symmetry beyond translation is a centre of inversion: each flag has a copy turned half round and moved to the underside, drawn open, and the dots are the centres. Every single cut leaves a chiral edge, and the piece on the far side carries its mirror image. The upper ribbon is cut symmetrically about a row of centres, so each centre on its middle line carries the ribbon onto itself with its two edges exchanged, and the ribbon is achiral. The lower ribbon, cut at the same width about a line with no centre on it, keeps nothing but translations and is chiral.

A ribbon can be achiral where its edges are not

Cut an achiral sheet once and the edge is usually chiral, in almost every direction. Cut it twice, into a ribbon, and the ribbon may be achiral even when both its edges are chiral, because an operation can carry the ribbon onto itself by exchanging the edges — but only if the ribbon's middle line sits on a centre of inversion or a mirror parallel to the cut. Seven of the eight sheets with no achiral edge give achiral ribbons, and one achiral sheet gives nothing but chiral strips however it is cut.

space-groups · Chirality

Named alongside it

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

ChiralityInversion centreLayer groupDeterminantFrieze groupHandednessHomomorphismRotoinversionContinuous symmetry measureCrystal classEnantiomorphFinite group

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