Crystallography & Symmetry
patterns derived from their groups
Concept

Axial vector — where it appears

A vector that picks up the determinant of an operation as well as its matrix, so that a mirror containing it reverses it and the mirror across it does not. A magnetic field is one, which is the whole reason its symmetry is a turning cylinder rather than a cone.

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

Also named here as curie principle, polar vector — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Crystal classCurie principleLimiting groupPermissionPolar vectorAnisotropyClosureDomain stateFerroelectricityInfinite groupIntersectionNeumann principle

All concepts