Crystallography & Symmetry
patterns derived from their groups
Concept

Orbit — where it appears

The set of places a group sends one point, which for a wallpaper group is the pattern itself. A pattern is not designed and then found to have symmetry; it is the orbit of a mark, and once that is taken literally it can be grown and checked.

Named by 31 essays across 8 fields — each of them below, with the objects they name alongside it.

A rotation. The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies.

What a symmetry actually is

Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.

operations · What symmetry is {111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone.

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

applied · Forms Growing the p4 orbit. One motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.

The orbit is the pattern

A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.

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 A fundamental domain for p4m. One representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

The fundamental domain

The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.

operations · Fundamental domain The general positions of P2₁/c. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. 12 general positions, the orbit of a three-point asymmetric motif, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

One part in however many, and why it is never quite that

A crystal's contents are the asymmetric unit repeated by the group. The unit's volume is the cell's divided by the order of the group — except that it is always a little more, and the excess is exactly the special positions counted whole.

operations · Fundamental domain p2 in one cell, p4 on average. On the left, a molecule in one orientation at a site whose symmetry is larger than its own: the arrangement has 2 operations and the detector says p2. On the right, the average over the 2 orientations the site offers, which is what a diffraction experiment measures because different cells choose differently and nothing prefers one choice. The average has 4 operations — it is p4 — and every atom in it is present in half of the cells. Both groups are detected from the point sets rather than assumed, and the difference between them is the reason a refined structure can have symmetry no molecule in the crystal has.

The symmetry of an average

A diffraction experiment measures an average over some 10²⁰ unit cells, and the average of several orientations is more symmetric than any of them. So a refined structure can carry symmetry that no molecule in the crystal has — including, in the worst case, a centre of inversion in a crystal built entirely of one hand.

restriction · Local symmetry 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 What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

operations · Counting How much a count of descriptions over-counts. For each plane group that has any two-colouring at all: how many colourings it has, how many designs those come to, and the ratio between them. Over the seventeen the ratio is 1.61, and group by group it runs from 1.00 — where nothing is identified — to 3.50 at p2, whose seven colourings fall into one class of six and one of one. The tick on each row is that row's largest single class, and it is at least the bar and usually more. The largest class anywhere is p2's 6, and that same group over-counts by only 3.50, because a factor is a mean over the group's classes and a mean reaches its largest term only when every term equals it. Reading the largest class as the over-count is therefore an over-statement, always. And the factor varies from group to group, which is why no single correction turns a count of descriptions into a count of designs after the fact.

Seventy-four colourings, forty-six groups

This site counts the two-colourings of the seventeen and gets seventy-four. The literature says there are forty-six two-colour wallpaper groups. Both numbers are right, and the gap between them is a disagreement about when two coloured patterns are the same pattern.

classification · Colour The most of an icosahedron a crystal can keep. Every subgroup of the sixty rotations of an icosahedron, found by closure, with the crystallographic ones marked — those whose rotation orders are all among the 1, 2, 3, 4 and 6 that a three-dimensional lattice admits. The largest is 23, of order 12, at index 5; everything containing a fivefold axis is refused. So a crystal containing an icosahedral molecule may fix a twelfth of the molecule's own symmetry and no more, and the remaining 5 orientations have to be related by something other than the site's symmetry.

The most of an icosahedron a crystal can keep

C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.

restriction · Local symmetry The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

lattices · Lengths What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

operations · Counting The cell of 3.4.6.4, and the vertices in it. 3.4.6.4 drawn with the cell its own translations define. The lattice is hexagonal and the cell holds 6 vertexes, marked. Neither was chosen: the translations are the vertex-to-vertex vectors that carry every polygon of the patch onto a polygon of the patch, and the cell is the shortest independent pair of them. Expressed in that basis the vertices have coordinates that are exact and are not fractions — a vertex of this tiling sits at 1/(1 + √3) of a cell — which is why the detector that decides its group works in ℚ(√3) rather than in the rationals.

Eleven tilings, five groups

Hand each of the eleven uniform tilings to a detector that has never heard of tilings and ask what its symmetry is. Six of them answer p6m. Twelve of the seventeen wallpaper groups never appear at all — and the coordinates the question has to be asked in are not fractions.

classification · Tilings Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs.

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

restriction · Finite groups 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 Dropping one invariant of 3m makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 3m's invariants takes on it — the 2 generators with the first one removed, over a window of 4 cells. With the full set, the 25 orbits of the group take 25 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work.

An orbit is what the invariants cannot tell apart

Two points of the plane lie in the same orbit of a group exactly when every invariant polynomial takes the same value on both. One direction of that is a definition; the other is a theorem, and it is checked here by comparing every pair of points in a window both ways.

operations · Counting anisohedral: 2 orbits of congruent tiles. A tiling of the plane by 8 copies of one shape per cell of a lattice of index 64, drawn 1 cell across and 8 up, and coloured by which orbit of the tiling's own symmetry group each tile belongs to. The group has 4 operations per cell and 2 orbits: every tile is congruent to every other, and no motion of the whole pattern carries a tile of one colour to a tile of another. Congruence is a fact about the shapes; an orbit is a fact about the pattern, and they are different facts.

One shape, two kinds of tile

A tiling by copies of a single shape looks as though it must be homogeneous — every tile is congruent to every other, so what could distinguish them? The symmetry group can. There are shapes that tile the plane and admit no tiling whose group carries any tile to any other, and the smallest of them has eight cells.

classification · Isohedral square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here.

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

applied · Defects An orbit of 12 points under 6mm, split into 6 kinds. The functions defined on one orbit of 12 points under 6mm form a space of that dimension, and the group acts on it by permuting the points. The character of that action is the cheapest one in the subject — at each operation it is the number of points left where they are — and decomposing it against the table gives the multiplicities on the right. They are whole numbers, they weight the dimensions to 12 again, and the multiplicity of the trivial representation is the number of orbits, which is Burnside's lemma arriving as a special case.

The shell that splits into kinds

The neighbours of an atom carry a space of functions as large as the shell, and the group does not treat that space as one thing. It splits into pieces of a few kinds, in whole numbers, and the count is the cheapest character in the subject: how many neighbours each operation leaves where they are.

point-groups · Representations What each Laue class buys, in measurements per reflection. The eleven Laue classes, with how many distinct reflections a block of indices holds under each and how many times a data set measures the average one. The redundancy is always below the order of the class and the gap is the special reflections. This is the number an experiment is planned around: repeated measurements of what symmetry says must agree are the only estimate of precision that does not come from a model, so a triclinic crystal has to be turned through far more of the sphere than a cubic one to be measured as well.

Every reflection, several times over

A diffraction experiment does not measure each reflection once. Symmetry relates a reflection to the others of its orbit, and those are the same reflection seen from another direction — so a hundred thousand measurements may contain twelve thousand reflections, each observed eight times.

diffraction · Resolution The parity argument loses 36 pairs it had won alone. The argument that refutes ten of the twenty-one species walks round a polygon of odd size: the ring of polygons about it is a closed walk of odd length in a graph the species decides, and a bipartite graph has no such walk. With two species at a vertex the flanking pairs come from the union of two graphs, and a union of bipartite graphs need not be bipartite — so the walk stops being constrained. The fourth row is the cost: pairs whose members the argument kills on their own and which it cannot kill together.

The argument that closes eleven

Twenty-one vertex species satisfy the angle equation; a parity argument kills ten before anything is drawn, and the eleven survivors are all built. Asking the same question of tilings with two kinds of vertex, the parity argument evaporates — it constrains a walk in a graph one species decides, and two species decide the union of two graphs, which need not be bipartite. What is left is a search, and a search cannot close a count.

classification · Decidability One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has.

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically — that is the first rung of this ladder and it is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

classification · Colour The table of marks of 4mm. Every conjugacy class of subgroup of 4mm, against every other. The entry is the number of cosets of the column's subgroup that the row's subgroup holds still. The first row is the identity, which fixes everything, so it is the size of each coset space; the last column is the whole group, whose only coset is fixed by everybody.

The table that decides every action

Burnside's lemma counts orbits and stops there — two completely different actions with the same orbit count are indistinguishable to it. The object that settles the whole question is a square table whose entries count fixed cosets: lower triangular because a subgroup fixes no coset of anything smaller, positive on the diagonal because it fixes its own, and therefore invertible. Inverting it turns a list of fixed-point counts back into the orbits themselves.

point-groups · Counting Disorder models against occupancies, site by site. Every distinct site symmetry in the space groups built here — 19 of them — with its order, its number of subgroups, the number of distinct disorder models, which are the subgroups up to conjugacy by the operations of the site symmetry, and the number of different occupancies those models can have. The last column is the largest number of models that share one occupancy. In all, 162 models share far fewer occupancies; the most crowded is 4/mmm, where 11 different models all give an occupancy of 1/4. At every site, the classes of operation a model keeps separate it from every other model with the same occupancy.

The occupancy does not name the disorder

A molecule disordered on a special position takes a number of orientations fixed by a group index, and its occupancy is the reciprocal. Many different disorders share one occupancy — eleven at a single kind of tetragonal site — and what separates them is which of the site's operations the molecule keeps, which the averaged structure records and the occupancy does not.

restriction · Local symmetry 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 The average is the site's orbit, with occupancies. A molecule at a site of symmetry mmm keeping a subgroup of order two takes four orientations, and the average over them is the site group's orbit of each of the molecule's atoms, every image at one over the length of its own orbit. Atoms in general positions give eight images at an eighth each, and give the same eight whichever subgroup the molecule keeps. Atoms on a locus the model keeps give a shorter orbit at a higher occupancy, drawn larger and darker, and those are the only atoms that differ between models. The total scattering is the same for every model, so all of them agree exactly at zero scattering angle.

The molecule size that hides a disorder

Two disorder models with the same occupancy leave averaged structures that differ only in a handful of partial atoms. The difference is 20% in structure factors for a ten-atom molecule and 3% for a sixty-atom one — so the data choose between the models for a small molecule and stop choosing for a large one, and seven pairs are identical at any size.

restriction · Local symmetry Two symmetric structures a diffraction pattern cannot separate. Two arrangements of 6 atoms on a 6 × 6 torus, each invariant under the plane group p6m, drawn beside the Patterson they share. No translation and no inversion carries one onto the other, so they are different structures; every one of the thirty-six interatomic vector counts is the same, so every diffracted intensity is the same and no measurement at any resolution separates them. Of the 4 structures with this symmetry and this many atoms, there are only 3 Pattersons — so imposing the most symmetric of the seventeen plane groups has not removed the ambiguity.

Symmetry does not rescue a Patterson

Every homometric pair found so far sits on a bare ring with no operations imposed, and a real crystal sits in a space group. Impose one and the ambiguity does not go away: 12 of the 13 groups searched still have pairs, and at six atoms the hexagonal groups are indistinguishable two to three times as often as the general position.

diffraction · Homometry Three conditions, and a near-miss for each. Zassenhaus's characterisation asks a group for a normal subgroup that is free abelian of finite rank, of finite index, and maximal among the group's abelian subgroups. Four groups against those three clauses. The free group on two letters has no non-trivial abelian normal subgroup at all; the discrete Heisenberg group has one that is free abelian of rank two and maximal abelian, and its index is infinite; ℤ² × ℤ/2 has a free abelian normal subgroup of index two, and the maximal one has torsion in it. Each fails a different clause, which is what shows no clause is redundant. The infinite dihedral group passes and is crystallographic in one dimension.

Which groups a crystal could have

Bieberbach's theorem is a statement about a group acting: discrete, no point far from an orbit. Zassenhaus turned it round into a statement a group can satisfy on its own — a maximal abelian normal subgroup, free of finite rank, of finite index — and each of those three clauses is kept out of redundancy by a group that fails it and nothing else.

restriction · Finiteness An orbit on a parabola, discrete and cocompact. The images of the origin under the group generated by two commuting affine maps of the plane: A slides one step along x and lifts y by the x it started at plus a half, and B is the translation by one in y. The images are the points with whole-number first coordinate and second coordinate a whole number above half the square of it, so the large dots lie on the dashed parabola and the small ones are the rest of the orbit. No two distinct images come closer than 1.000, and no point of the square between the axes lies farther than 0.610 from one — so the action is discrete and its quotient is compact, which is exactly what Bieberbach's first theorem asks for.

Straight lines, and no distances

Every finiteness met so far rests on the motions preserving a metric, because the trick that produces one is an average and an average needs something to average over. Keep the straight lines and drop the distances, and Bieberbach's first theorem is false in the plane — by an example two lines long, whose group is the plane's own translations and whose translations have rank one.

restriction · Finiteness

Named alongside it

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

StabiliserSpecial positionMultiplicitySite symmetryEnumerationConjugationFinite groupFixed pointHolohedryNormaliserPlane groupSubgroup

All concepts