Crystallography & Symmetry
patterns derived from their groups
Concept

Torus — where it appears

The surface obtained by gluing the opposite edges of a parallelogram, which is what a finite periodic block of a crystal is. Its translation group is finite and its characters are the only wavevectors such a block can carry.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

classification · Flat space 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.

classification · Flat space Two plane structures with one Patterson. Two arrangements of 4 atoms on a 4 by 4 torus. No translation and no half-turn carries one onto the other, so they are different structures; every interatomic vector occurs the same number of times in both, so no measurement of intensities distinguishes them. The plane case is not the chain case with an extra index: a mirror in the plane sends a vector set to its mirror image rather than to itself, which is the one place the analogy with a cycle breaks.

Two structures on a torus, and one Patterson

Homometry was settled here on a ring of positions, which is a crystal in one dimension. Moving the same exhaustive search to a torus asks whether the coincidence is commoner or rarer when the vectors have a plane to land in — and the honest answer is that dimension is not what decides it.

diffraction · Homometry The sphere fixes a count; the torus fixes only a difference. Euler's relation for a trivalent net gives Σ (6 − n) pₙ = 6χ, so the surface fixes one linear combination of the face counts and nothing else. On a sphere that combination is twelve, which with no face smaller than a pentagon forces exactly twelve pentagons. On a torus it is zero, which permits any number of pentagons provided as many heptagons pay for them — and permits none at all, which is the plain hexagonal net. On a surface of two holes it is minus twelve, so heptagons become compulsory instead.

As many heptagons as pentagons

A trivalent net on a sphere must have exactly twelve pentagons. The same three lines of arithmetic on a torus give zero — which does not forbid pentagons, it makes them pay: every pentagon has to be balanced by a heptagon, and the counts are otherwise free. One rotated bond in a wrapped honeycomb makes two of each and changes nothing else.

restriction · Curvature Every closed surface, and the two that charge nothing. The same accounting indexed by Euler characteristic rather than by genus. An orientable surface has χ = 2 − 2g, so it only ever occupies an even row; a non-orientable one has χ = 2 − k and occupies every row from one downwards. The odd rows therefore belong to surfaces that cannot be oriented and to nothing else — and the first of them, the projective plane, charges six. Six pentagons is a bill no orientable surface presents.

The surfaces a count by genus skips

A count indexed by genus steps in twelves and lands only on even numbers. A closed surface can have any characteristic at or below two, and the odd ones belong to the surfaces that cannot be oriented — where the projective plane charges six pentagons, a bill no orientable surface ever presents.

restriction · Curvature

Named alongside it

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

Crystal netCombinatorial curvatureThe Euler characteristicFree actionOrientabilityQuotient graphBurnside lemmaChinese remainder theoremCommensurateCoordination numberCrystallographic restrictionDefect

All concepts