Crystallography & Symmetry
patterns derived from their groups
Concept

Change of basis — where it appears

A relabelling of the vectors describing a lattice, which alters every coordinate and no physical fact. The changes available are exactly the integer matrices of determinant plus or minus one, and in the plane they are all words in two moves.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

One group, three symbols

P2₁/c, P2₁/a and P2₁/n are the same space group written on three choices of axes, and the literature contains all three as though they were different. Deriving a symbol from a group's own operations shows why — and found an entry on this site that had been carried under another setting's name for two phases.

space-groups · Settings 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 The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

The space every lattice lives in

Five lattices in the plane is the number of *kinds*. The number of lattices is a continuum — and it has a shape: one two-dimensional region with two corners, three edges and an interior, where the five kinds turn out to be a region, three arcs and two points rather than five things of one sort.

lattices · Moduli The region, and its copies. Words in S and T up to length 4, each carrying the region somewhere else. The copies do not overlap and they do not leave gaps: the upper half-plane is tiled by them, one copy per change of basis. That is the whole content of the claim that reduction picks a canonical basis — every basis of every lattice is in exactly one copy, and reduction is the walk back to the shaded one.

Two moves reach every basis

A lattice has infinitely many bases and reduction picks one. Why it can is a fact about a group with two generators and two relations — and the fundamental region tiles the plane with its own copies, one per basis, which is what makes the walk home finite.

lattices · Moduli p4: the map comes back. p4 written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found.

The same group means the same pattern

Seventeen patterns is not the same statement as seventeen groups. Two patterns that look nothing alike could in principle have symmetry groups that are abstractly the same, and then the classification would be a classification of drawings. Bieberbach's theorem says they cannot — and the affine map that proves it can be recovered from the two operation sets alone.

restriction · Finiteness the honeycomb net, unfolded over 3×3 cells. The infinite graph the quotient graph names, drawn over 3 by 3 cells with the home cell outlined. Each edge of the quotient becomes one edge per cell, running to the cell its voltage names; the drawing adds coordinates the net does not have, and they are the placement in which every vertex sits at the average of its neighbours. two vertices, three edges, degree three — the graph of graphene and of every hexagonal mesh.

A structure with the distances thrown away

Keep which atoms are joined and throw away where they are, and what is left is an infinite graph that can be written on a postcard: a few vertices, a few edges, and a pair of integers on each. Two things about that writing-down are free, and neither of them changes the net.

applied · Nets One vertex, two edges: one net. Three edges: no answer at all. Every net with one vertex and the stated number of edges, counted inside boxes of voltages of three sizes, up to change of basis and the sign of an edge. Two edges give one net whatever the box, and the reason is a sentence: two voltages that generate the translations are a basis of ℤ², and every basis is carried to every other. Three edges give more nets in every larger box, and that is not a failure of the search — normalise two of the voltages to a basis and the third is a free pair of integers, so the family is infinite. An enumeration inside a bound reports which of those two situations it is in rather than reporting the count it happened to reach.

Every net with one vertex, counted

A net is a few vertices, a few edges and a pair of integers on each, so a census is available: fix the numbers, bound the integers, enumerate. Two edges give exactly one net at every bound. Three give three, then nineteen, then a hundred and forty-three — and the question changes.

applied · Nets One net, six descriptions, four different answers about its symmetry. The honeycomb written against six bases of ℤ², all of them the same net. The detector tests each lattice type's holohedry in standard position, so a symmetry written against another basis is a matrix that is not in the list and is never tried — and the answer comes back as p6m, or an unnamed group of order four, or p2, or cmm, depending on how the voltages were typed. The metric column is the form the net's own edges make, inverted; the reduced column is that form after Lagrange–Gauss reduction, and it is the same in every row, which is what makes the last column a property of the net.

The symmetry a net was written with

A net has no coordinates, so its symmetry is whatever its best drawing has. This collection measured that by handing the drawing to a detector — and the detector tests a fixed list of matrices, so the answer depended on which pair of translations the voltages had been written against. The honeycomb came back as p6m, or p2, or cmm, or nothing, one net and four answers.

applied · Nets Two vertices and three edges: two nets, at every box size tried. Every net with two quotient vertices and the stated number of edges, counted inside boxes of voltages of several sizes. One cross voltage is set to zero by the gauge — the freedom that moving one vertex into another cell gives — and the rest are drawn from the box. Each entry is the count of nets whose placement separates their vertices, plus the count of those whose does not: the first has a canonical description and stops growing, and the second does not have one and therefore keeps rising with the box. The reducible column is the descriptions thrown away for a reason the one-vertex census never had — cycles generating the whole of ℤ² and a net whose own cell holds one vertex rather than two — and it is empty at every odd edge count, because the swap that would reduce a description pairs its edges and an odd number cannot pair.

Every net with two vertices, counted

The one-vertex census could not contain the honeycomb, because the honeycomb has two vertices in its cell. Adding the second one closes a family at two nets, removes the floor of p2 entirely, makes a third of the members undrawable, and forces the census to refuse a kind of description the first one never met: an honest quotient graph written on twice the cell it needs.

applied · Nets One change of basis turns a Gram into its own adjugate. For each Gram matrix: the matrix after the basis change by a right-angle rotation, and the adjugate. They are equal, always — and the adjugate is the determinant times the inverse, which is the dual lattice's Gram. So the dual is the same lattice on a rotated basis, scaled by one over the determinant. Five rows are the named plane lattice types and the rest have entries picked at random, because the claim is an identity in integers and not a property of the five.

Every plane lattice is its own dual

The dual of a lattice has the inverse Gram matrix, and in two dimensions the inverse is the adjugate over the determinant — which is what one particular change of basis does to a Gram. So a plane lattice's dual is the lattice itself, turned through a right angle and scaled, for every lattice with no exception. In three dimensions it is a condition, and the face-centred and body-centred cubic lattices are duals of each other rather than of themselves.

lattices · Lattice 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 One change of setting, four rules. The four things a structure report contains and the rule each obeys under a change of setting with basis change P and origin shift p. The cell and the indices are multiplied by P; a coordinate is multiplied by its inverse, after the origin has been subtracted; and an operation is conjugated and then shifted by (W − I)p, a term the other three have no equivalent of. Applied to Pnma with the change below, the operations still close into 8, the orbit maps point for point, and every |F| is unchanged.

One matrix, four rules

Changing the setting of a structure is one matrix and one origin shift — and the cell, the coordinates, the indices and the operations each obey a different rule under it. Three of the four ways of getting it wrong still leave a closed group of the right order, so closure catches none of them.

space-groups · Settings

Named alongside it

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

Crystal netQuadratic formQuotient graphUnimodular matrixVoltageBarycentric placementBasis reductionGraph isomorphismSettingBravais latticeCensusDeterminant

All concepts