Crystallography & Symmetry
patterns derived from their groups
Concept

Determinant — where it appears

The factor by which a linear map scales area or volume, negative when the map reverses handedness. For a symmetry operation it is +1 or −1, and which one it is decides whether a structure of a single enantiomer can have that operation.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

Pnma has 6 names. The group Pnma with its three axes relabelled in each of the six possible ways. Every row is the same group, and each row is checked as it is drawn: the change of basis has determinant one, conjugating back gives the original operations exactly, and the census of screws, glides, mirrors and rotations is unchanged — and 6 different symbols come out. Each symbol is derived from the conjugated operations, not looked up, by the same routine that has to reproduce the symbol every group was entered under.

Six ways to name one group

Pnma is also Pmnb, Pbnm, Pcmn, Pmcn and Pnam. Nothing about the crystal changes between those six; what changes is which axis was called a. In an orthorhombic group the axes are inequivalent and unlabelled, and naming them is a choice made six ways.

space-groups · Settings Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.

How many arrangements one rule allows

Every count in this collection so far has been a count of symmetries, or of orbits under one. Here is a different count: the arrangements a purely local rule permits on a fixed lattice, with no symmetry quotient anywhere in it. The answers are enormous, they are exact, and the useful quantity is not the number but its growth per site.

aperiodic · Entropy Perfection is a rank, and most lattices do not reach it. For each lattice, the rank of the matrices vvᵀ built from its shortest vectors, against the dimension of the space of symmetric matrices those live in. Reaching it means the shortest vectors pin the form down completely: no deformation keeps every one of them at its length. Falling short means there is a direction left to move in, and the lattice is not a local maximum of density.

One perfect form in space

Which lattice packs spheres most densely is a question about a maximum over a continuum, and Voronoi turned it into a rank calculation and a sign check. A lattice is a local maximum exactly when its shortest vectors pin its shape down completely and its inverse can be written over them with positive coefficients. Searching every reduced integer form of minimum two finds one such lattice in the plane and one in space.

lattices · Packing How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have.

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

lattices · Sublattices Thirty-two classes, from fourteen Gram matrices. The five hundred and ten subgroups sorted by how many operations of each kind they contain — a determinant and a trace decide which of the ten kinds a matrix is. Thirty-two answers come out, and they are the thirty-two crystal classes: matched against the construction elsewhere in this collection by signature rather than by name, since nothing here names a point group.

Thirty-two from fourteen matrices

Write the fourteen Bravais lattices as Gram matrices, ask each one which integer matrices preserve it, and take every subgroup of every answer: five hundred and ten of them. Sort those by how many operations of each kind they contain — which a determinant and a trace decide — and thirty-two answers come out. They are the crystal classes, from a construction in which no point group is ever named.

point-groups · The fourteen Bravais lattices Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 63 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet.

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

space-groups · Chirality A thread's two signs, and the four kinds of operation. Every operation of a rod group carries the axis to itself, so it does two independent things: it keeps or reverses the direction along the thread, by a sign σ, and it keeps or reverses the handedness of the plane across the thread, by the determinant of a 2 × 2 matrix. The determinant in space is the product, so the shaded cells are the proper operations — a turn or screw about the axis, and a half-turn crossing it — and the unshaded ones are the improper. A rod group is chiral when all of its operations sit on the shaded diagonal, and polar along its axis when all of them sit on the top row. The two conditions pick out different diagonals of the same square, which is why neither implies the other.

A thread's hand is not a choice

A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.

space-groups · Chirality

Named alongside it

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

Bravais latticeGram matrixChange of basisChiralityCrystal classEnumerationHomomorphismImproper operationLatticePoint groupQuadratic formSohncke group

All concepts