Crystallography & Symmetry
patterns derived from their groups
Series

Flat space — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

    The two that fold into a surface

    Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

    part 1 · classification
  2. Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

    Ten ways for space to be flat

    Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

    part 2 · space-groups
  3. 11 nets, and one accounting. Every plane net folds onto a torus when its own translations are divided out, and a torus has Euler characteristic zero — so the quotient's vertices, edges and faces satisfy n − e + f = 0 and the number of faces is not something to be counted off a drawing but e − n. Dividing through gives one over the mean face size plus one over the mean degree equal to a half, which is the same relation that forbids a plane tiling by pentagons, reached here with no geometry in it at all. It holds for every net in the table.

    Every net folds onto a torus

    Divide a plane net by its own translations and the quotient is a finite graph drawn on a doughnut. A doughnut has Euler characteristic zero, so the number of faces is not something to count — it is forced, and with it a relation between how many edges meet at a vertex and how many bound a face.

    part 3 · classification
  4. A wavevector of thirds, and the boxes that cannot see it. Which sizes of box can carry the wavevector at the corner of a hexagonal zone. The characters of the box's translation group are its wavevectors, and there are exactly N² of them — the fractions with denominator dividing N. A wavevector of thirds is therefore present in a box of three, six, nine or twelve cells and absent from one of two, four or five: not approximated badly, not resolved coarsely, absent. A mechanism or a level living there is invisible to such a calculation, and that is the practical content of a mechanism count depending on the cell it was looked for in.

    Crystallography in a box

    A calculation over a crystal is not performed on a crystal. It is performed on a finite block with its edges glued, and the block has a symmetry group of its own — finite, complete in one direction and missing something decisive in the other.

    part 4 · classification
  5. One arm of a star is cheap, and the whole star costs the cube. For nine wavevectors of the primitive cubic zone, the number of cells in the smallest periodic box that holds one arm of the wavevector's star, and in the smallest that holds every arm, on a logarithmic axis found by enumerating every box of each size. The zone corner R is held whole by two cells. The face centre X needs two cells for one arm and eight for its three; the thirds along an axis or a body diagonal need three for one arm and twenty-seven for the star; the quarters need four and up to sixty-four. Every smallest box for a whole star has the full symmetry of the cube.

    A whole star needs a cubic box

    A periodic box for a cubic crystal holds only the wavevectors its own lattice allows, and a box chosen to hold one wavevector usually holds one arm of its star and not the others. One arm of the face centre X fits in two cells; all three need eight. One arm of a third along an axis fits in three; the whole star needs twenty-seven. The smallest box that holds a whole star always keeps the cube's full symmetry, so a calculation either pays for the cube or reports a symmetry its box imposed.

    part 5 · classification
  6. Whole stars of the hexagonal zone, and the boxes that hold them. For ten wavevectors of the primitive hexagonal zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it differs, in the smallest n by n grid of cells holding the star. The corner K is a single arm up to sign and is held whole by three cells, against nine for a grid; K raised to the top face, H, by six against eighteen; halfway to K, T, by twelve against thirty-six. M needs two cells for one arm and four for its three, and L two and eight. Every smallest box for a whole star keeps the full hexagonal symmetry.

    The corner of a hexagon fits in three cells

    A periodic box for a hexagonal crystal holds a whole star of wavevectors only if it keeps the crystal's full symmetry, as a cubic box must. What the cube never offered is a box turned through thirty degrees. It holds the corner of the hexagonal zone in three cells where every grid needs nine, and the saving is always exactly a factor of three and always at the same stars. Turn a box through any other angle that the six-fold axis allows and it loses the mirrors, and comes with a mirror image.

    part 6 · classification
  7. Whole stars of the rhombohedral zone, and the boxes that hold them. For eight wavevectors of the rhombohedral zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it costs more, in the better of the two regular grids: the rhombohedral cell repeated n times along each edge, or the hexagonal triple cell repeated n by n by m. The axis end T is held whole in two cells against six, the face centre F in four against eight, and the in-plane third K₃ in nine against twenty-seven; L needs eight either way. Every smallest box for a whole star keeps all twelve operations of 3̅m.

    A rhombohedral box saves a factor of two the hexagon never offered

    A rhombohedral crystal is drawn on two cells, the small rhombohedron and the hexagonal cell three times its size, and a periodic box is usually one of them repeated. The smallest box that holds a whole star of wavevectors is the star's own lattice instead, and across every star with denominators up to eight it beats the better of the two grids by one, two or three, never more. Three was the hexagon's number. Two is new, and it is the face centre's. And the boxes that keep the three-fold axis and lose the mirrors still come in mirror pairs, now at nine cells as well.

    part 7 · classification

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