Crystallography & Symmetry
patterns derived from their groups
Concept

Equivalent positions — where it appears

The full orbit of a general point under a space group, listed as coordinate triples. It is the form in which the Tables state a group, and every coordinate list a crystallographer publishes is one representative per orbit.

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

A conjugacy class of p4m. One conjugacy class of p4m drawn in place: every copy of the same symmetry that the group can carry onto every other. Conjugation was applied to each of the 8 operations by each of them in turn, and the kind and order of the result was checked to match every time.

The same symmetry, somewhere else

Two mirrors in a pattern can be the same symmetry or two different ones, and looking will not settle it. Conjugation is the operation that decides, and it turns an intuition about sameness into arithmetic.

operations · What symmetry is Special positions in p4m. Every point of a 12×12 grid inside the cell of p4m, drawn at a size set by how many operations fix it. 80 of the 144 are general — nothing but the identity leaves them alone, so their orbit is the full 8 points. The other 64 are special, and fall into 3 kinds: 60 points fixed by 2 operations, with orbits of 4; 2 points fixed by 4 operations, with orbits of 2; 2 points fixed by 8 operations, with orbits of 1.

The points a group treats differently

Almost every point of a cell has an orbit as long as the group. The exceptions are the points some operation leaves alone, and they are where atoms sit, where a structure's formula comes from, and where a careless motif destroys the group it was meant to illustrate.

operations · Fundamental domain

Named alongside it

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

Site symmetryConjugacy classConjugationMultiplicityNormaliserOrbitSpecial positionStabiliserWyckoff positions

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