Crystallography & Symmetry
patterns derived from their groups
Concept

Involution — where it appears

An operation that undoes itself, so that doing it twice is doing nothing: a mirror, a half-turn, an inversion. In a pattern's group the involutions are the operations of order two, and a motion that is a product of two of them can be run backwards by one of them.

Named by 2 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 Which plane groups can run every operation backwards, kind by kind. For each of the seventeen plane groups and each kind of operation it has, whether every operation of that kind in a window of 5 × 5 cells is conjugate in the group to its own inverse, decided exactly. The groups in which every kind passes, shaded, are p2, pmm, cmm, p4m, p6m. The last column is the answer the point group gives, which is yes for fourteen; it is wrong for nine of them.

Five groups can run every motion backwards

Every symmetry of a pattern has an inverse, the same motion run backwards. Sometimes the pattern's own group contains a third motion that turns the one into the other, so that a motion and its reverse are the same symmetry seen from somewhere else. In five of the seventeen plane groups that is true of every operation, and the group's point group, which is where anyone would look, says it is true of fourteen. The difference is nine groups, and every one of them fails on something the point group cannot see: a translation, a glide's slide, or a rotation centre with no mirror through it.

operations · What symmetry is

Named alongside it

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

ChiralityConjugacyConjugacy classContinuous symmetry measureGlide reflectionHandednessImproper operationPoint groupWallpaper group

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