Crystallography & Symmetry
patterns derived from their groups
Concept

Conjugation — where it appears

Applying an operation, then a second, then undoing the first — the arithmetic deciding when two symmetries are one symmetry differently placed. It is also the operation under which a group's own normaliser is defined, one level up.

Named by 12 essays across 3 fields — 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 The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

The same pattern, described twice

Two coordinate lists for one structure can disagree in every number and describe exactly the same arrangement, because a group does not fix its own origin. The operations that may be applied to a description without changing what it describes are its normaliser, and they can be found by looking at pictures rather than at matrices.

operations · Normalisers 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 Averaging a metric over the group. The 3 pale ellipses are the unit circle carried by each element of a finite group of rational matrices — none of them a rotation, because the group has been skewed out of the orthogonal ones on purpose. Their average is the heavy ellipse, and it is invariant: MᵀAM = A for every element, exactly, in rational arithmetic. So a finite group of matrices is always a group of isometries of some inner product, and every question about how large such a group can be becomes a question about the symmetries of an ellipse. The space of invariant forms here is 1-dimensional, so up to scale the average is the only one.

The average that makes it finite

Two arguments every classification leans on are usually assumed rather than made: that a finite group of motions fixes a point, and that a finite group of integer matrices preserves a metric. They are the same trick — average over the group — and the trick fails exactly where it should.

restriction · Finiteness Every subgroup of index two is normal; at index three most are not. For each of the seventeen plane groups, its abelianisation and the number of normal subgroups of each small index against the number of subgroups of that index. The index-two column is complete every time, because the left and right cosets of a subgroup of index two are the same pair of sets. At index three and four the two numbers part, and the gap is what normality costs: a subgroup that is carried to a different subgroup by some operation of the group it sits in.

The quotient each normal subgroup leaves

Two hundred and eighty-one subgroups of index four across the seventeen plane groups, and ninety-seven of them normal. Which ones, and what is left when they are divided out, needs no enumeration at all: below order six every group is abelian, so a normal subgroup of small index is a subgroup of the abelianisation and its quotient is decided by a product of greatest common divisors.

operations · Subgroups Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero.

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

operations · Composition Four groups whose description count a metric can raise. Every plane group, the number of ways of writing one arrangement down on the lattice the group requires, and the number on the most symmetric lattice it may sit on. Eight groups already occupy the most symmetric lattice available to them and have nowhere to go. Four have a metric that raises the count, by two and in one case by six. The starred rows belong to groups whose normaliser has a free direction, where the quantity is a count of grid points rather than an index and cannot be compared.

The normaliser is not a function of the group

How many ways there are of writing one structure down is computed from the group and printed in a table beside its name. It is not a property of the group. Draw a p2 pattern on a hexagonal cell rather than an oblique one and the number goes from four to twenty-four, with nothing done to the group at all.

operations · Normalisers One achiral motif in p4, chiral along one line and achiral along two. The same motif — three points with a mirror and no other symmetry — repeated by p4, the plane group of quarter-turns with no mirror, and placed with its mirror along three different lines of the square lattice. Along the first line the pattern has no operation that reverses orientation: it is chiral, although every piece of it is achiral. Along the second and third the motif's mirror is also a mirror of the whole pattern, and the detected groups are p4m and p4g; the mirror lines of each pattern are drawn. All three verdicts come from detecting the symmetry of the points and agree with whether the motif's mirror normalises p4.

A hand made of pieces that have none

Quartz is built from tetrahedra that have no handedness, and every quartz crystal is left-handed or right-handed anyway. Put a piece with a mirror into a pattern whose group has none, and the pattern keeps the piece's mirror only if that mirror lies on one of a few lines the group's normaliser draws. Anywhere else, the arrangement has a hand its parts do not.

space-groups · Chirality Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm.

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

operations · Normalisers p2's symmetries, sorted into classes by the group itself. A pattern with the symmetry of p2 over 2 by 2 cells, with its rotation centres and mirror lines marked in the International Tables' shapes and coloured by conjugacy class in the infinite group: two marks share a colour exactly when some operation of the group carries one element onto the other. Where rotations of several orders share a centre, the mark is the highest order's and so is its colour. Glides are not drawn. Classes counted: half-turns: 1 in the quotient, 4 in the group.

Two mirrors a coset cannot tell apart

Taken modulo its lattice a wallpaper group is finite, and its conjugacy classes are easy to list. But a coset holds every mirror of one direction at once, and the group itself keeps apart mirrors the list merges: pm has two classes of mirror, p2 four classes of half-turn, p3 six classes of rotation. Deciding which is which is Dehn's conjugacy problem, and for these groups it comes down to whether one vector lies in one lattice.

operations · What symmetry is Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points.

Three of them, and they are equivalent

The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.

operations · Normalisers One lattice, two copies of pm. pm on a lattice doubled across its mirrors. The mirrors of the parent are every vertical line; a copy of pm on the doubled lattice has mirrors every other line, and there are two ways to choose which — the solid set or the dashed set. Both are copies of pm with the same lattice and the same point group, and no translation of the parent carries one onto the other, because the translation that would is exactly the one the doubling removed. So a count of invariant lattices is not a count of subgroups, and the gap is visible in the smallest case there is.

A lattice is not a subgroup

Every count of copies so far has counted lattices, and the International Tables count subgroups. One invariant lattice can carry several copies of a group that nothing in the parent carries onto one another — pm's doubled lattice carries two, with its mirrors on the even lines or the odd ones — and how many is a cohomology computation, a first where the classification of the seventeen used a second.

space-groups · Isomorphic subgroups

Named alongside it

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

NormaliserIndexOrbitConjugacy classEquivalent originHolohedryMetric tensorNormal subgroupSite symmetryAbelianisationCosetFixed point

All concepts