Crystallography & Symmetry
patterns derived from their groups
Concept

Permission — where it appears

A statement that symmetry allows something, as opposed to a prediction that it happens. Almost every result in this collection's applied fields is one, and reading a permission as a value is the commonest way of misusing a symmetry argument.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

The seven groups a field can have

Every group in this collection so far has been finite, because a lattice forbids the alternatives. A uniform field has no lattice: rotate it about its own axis through any angle at all and nothing has changed. There are exactly seven such groups, and they come out of the same closure argument that turns sixteen frieze candidates into seven.

point-groups · Curie What a crystal keeps of itself in a field. Each class, with what is left of it when a field is applied along the axis of its own setting. The residual is the intersection of the class with the field's own group, computed on matrices and matched against the thirty-two rather than named by hand. Where the residual is the class itself, the field takes nothing away — and for an electric field those are exactly the polar classes.

What a crystal keeps in a field

Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.

point-groups · Curie Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other.

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

applied · Packing

Named alongside it

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

Axial vectorCrystal classCurie principleLimiting groupPolar vectorAnisotropyClose packingClosureDomain stateFerroelectricityGlide reflectionGroup frequency

All concepts