Crystallography & Symmetry
patterns derived from their groups
Concept

Equivalence — where it appears

The rule that decides when two objects count as the same, and therefore what a classification is a classification of. Getting it wrong is this collection's commonest shape of error: a computation that is right about an equivalence that is not the intended one.

Named by 4 essays across 4 fields — each of them below, with the objects they name alongside it.

How much a count of descriptions over-counts. For each plane group that has any two-colouring at all: how many colourings it has, how many designs those come to, and the ratio between them. Over the seventeen the ratio is 1.61, and group by group it runs from 1.00 — where nothing is identified — to 3.50 at p2, whose seven colourings fall into one class of six and one of one. The tick on each row is that row's largest single class, and it is at least the bar and usually more. The largest class anywhere is p2's 6, and that same group over-counts by only 3.50, because a factor is a mean over the group's classes and a mean reaches its largest term only when every term equals it. Reading the largest class as the over-count is therefore an over-statement, always. And the factor varies from group to group, which is why no single correction turns a count of descriptions into a count of designs after the fact.

Seventy-four colourings, forty-six groups

This site counts the two-colourings of the seventeen and gets seventy-four. The literature says there are forty-six two-colour wallpaper groups. Both numbers are right, and the gap between them is a disagreement about when two coloured patterns are the same pattern.

classification · Colour In the plane, the lengths do name the lattice. Every reduced binary form with coefficients up to 20 — 1750 lattices — with its theta series computed to 120 terms. No two of them agree. That is Schiemann's theorem for binary forms, which says the theta series determines the lattice in two dimensions and in three, confirmed here as far as the search reaches rather than proved. The closest pair is worth the space: two lattices whose shortest vectors both have squared length twenty agree for 38 terms — because neither has any vector before then — and part at the next one.

The lengths do not name the lattice

Seventeen hundred plane lattices, every one with a theta series shared with no other — the lengths determine the lattice, and an exhaustive search says so. In sixteen dimensions two different lattices have identical counts at every distance, and the example is sixty years old.

lattices · Lengths P222: 16 descriptions of one structure. P222 has 4 operations in a cell. 8 origins leave every one of them exactly where it was, and 8 linear parts of the lattice's holohedry normalise the group, so its Euclidean normaliser has 64 elements per cell and the index is 16. That index is the number of coordinate lists that describe one and the same arrangement of atoms. Each was applied to a motif and the resulting point sets compared: the numbers differ and the sets are identical, which is the check that makes the count mean anything.

One crystal, and sixteen coordinate lists

Two structure reports can disagree in every number and describe the same arrangement of atoms, because a space group does not fix its own origin or its own axes. How many genuinely different lists there are is the index of the group in its Euclidean normaliser — a number, computable, and the thing a structural database has to divide out before it can say two entries are the same compound.

operations · Normalisers Thirty-two classes, eighteen groups. Every abstract group the thirty-two crystal classes realise, with the classes that realise it. 8 of the eighteen carry more than one class, and the largest collision is the four hexagonal classes that are all the dihedral group of order twelve. Nothing here is looked up: two classes are put in the same row when a search over images of a generating set finds a bijection preserving multiplication, and the search is finite because a generating set is small and the elements it may map to are the ones of the same order.

Thirty-two classes, eighteen groups

An inversion centre, a mirror and a two-fold rotation are three of the most different things a crystal can have, and they are the same group of order two. Forget the matrices and keep the multiplication table, and the thirty-two classes collapse to eighteen.

point-groups · Crystal classes

Named alongside it

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

HomomorphismAbstract groupAffine normaliserBasis reductionCharacterClassificationColour groupConjugacyCosetCounterchangeCrystal classDecidability

All concepts