Crystallography & Symmetry
patterns derived from their groups
Series

Curie — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · point-groups
  2. 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.

    part 2 · point-groups
  3. How many constants a texture permits. Every one of Curie's seven groups against every property this collection computes, as the number of independent components each permits. The counts are averages of a character over an infinite group, which is exact because the character is a trigonometric polynomial: the average is its constant term. A poled ceramic is the row ∞m, with one pyroelectric coefficient, two dielectric constants, three piezoelectric moduli and five elastic ones — and a zero for optical activity, which the mirrors forbid.

    What a texture permits

    A poled ceramic has no lattice, no cell and no class, and yet the number of piezoelectric moduli it may have is exactly three. The group is one of Curie's, the average over it is an integral, and the integral turns out to be a single Fourier coefficient — which is why the answer is exact and why a texture is indistinguishable from a hexagonal crystal until rank six.

    part 3 · point-groups

All series