Crystallography & Symmetry
patterns derived from their groups
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The theme: The same arithmetic, renamed — page 2

A crystal form is an orbit. A twin law is a coset. The domain states left by a phase transition are the cosets of the low-symmetry group in the high-symmetry one. Four subjects that grew up in different centuries and different departments, doing one piece of arithmetic under four names.
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. Operations

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.

 Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it. Lattices

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

 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. The classification

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.

 Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation. How it is known

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

 Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused. Into space

The plane that carries two glides

A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop 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. Into space

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.

 Three shapes, and nothing else. The dielectric tensor of a crystal is an ellipsoid, and averaging a generic one over a point group leaves exactly three possibilities: a sphere, where all three principal values agree and the crystal is optically isotropic; a spheroid, where two agree and there is one optic axis; and a general ellipsoid, with two. The counts are 5, 19 and 8 of the thirty-two classes, and they were found by computing the eigenvalues rather than by sorting the classes by system. What symmetry decides

Three optical characters, and the arithmetic that assigns them

A cubic crystal cannot be birefringent, whatever it is made of. Between crossed polars it stays dark at every rotation, and the reason is that averaging any ellipsoid over a cubic point group leaves a sphere — a permission computed before anybody measures anything.

 Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off. Symmetry at work

Which faces a crystal shows

Rock salt grows as cubes, fluorite as octahedra, garnet as dodecahedra. All three have cubic lattices and the same list of possible faces, and what separates them is which reflections are systematically absent — a rule about diffraction predicting a shape a mineralogist can hold.

 p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group. Operations

A group in four letters

Every other essay here describes a symmetry group by what it does to the plane. There is a second description — a handful of letters and the words in them that are required to equal nothing — and it can be counted with no plane anywhere in the computation.

 Subgroups of index two, three and four. Every plane group with the number of subgroups it has at each small index, counted by enumerating the transitive actions on that many points. The zeros are the interesting entries: p3 has no subgroup of index two and the four-fold groups have none of index three, because a subgroup of index n gives an action on n points and the group has to have a quotient that can act. A rotation of order three has nowhere to go in a set of two, and one of order four has nowhere to go in a set of three that is not the identity — so the index is constrained by the point group before any geometry is done. Operations

How many subgroups of index three

Taking operations away and closing what is left finds the maximal subgroups and stops there. Counting instead the ways a group can act on three points finds all of them — and finds that a four-fold group has none of index three at all.

 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. Lattices

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.

 The ball of radius 5 in p6. Every element of p6 reachable in at most 5 multiplications by a generator or its inverse, plotted at its translation part — so each dot is a lattice position and its size says how few steps reach it. The picture is the word metric's unit ball scaled up, and its shape is what fixes the growth: a diamond where the group supplies two short translations, and a hexagon where it supplies three. Every dot here required the word problem to be solved, because the search has to know when two products are the same element. Operations

Telling two words apart

There are finitely presented groups in which no algorithm can decide whether two products of the generators are the same element. The seventeen are not among them, and the procedure that settles it is short enough to state in a sentence — which then makes it possible to measure how fast each group grows.

 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. Operations

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.

 In the plane, the lengths do name the lattice. Every reduced binary form with coefficients up to 20 — 1750 lattices — with its theta series computed to 120 terms. No two of them agree. That is Schiemann's theorem for binary forms, which says the theta series determines the lattice in two dimensions and in three, confirmed here as far as the search reaches rather than proved. The closest pair is worth the space: two lattices whose shortest vectors both have squared length twenty agree for 38 terms — because neither has any vector before then — and part at the next one. Lattices

The lengths do not name the lattice

Seventeen hundred plane lattices, every one with a theta series shared with no other — the lengths determine the lattice, and an exhaustive search says so. In sixteen dimensions two different lattices have identical counts at every distance, and the example is sixty years old.

 Modulo 3 injective on all thirteen, modulo 2 on 5. Minkowski's lemma says the kernel of reduction modulo an integer of at least three is torsion-free, so a finite group of integer matrices is carried faithfully into a finite group of matrices over ℤ/3 — which is why the classification is finite, before any bound is computed. The middle column checks it on every finite subgroup of GL(2,ℤ) there is: thirteen classes, no collapses. The right column is the case the lemma has to exclude. Modulo 2, minus the identity is the identity, and 8 classes lose operations. What a lattice forbids

Reduction modulo three

A finite group of integer matrices survives being reduced modulo three: no two of its operations collide. That single fact proves the classification finite without computing any bound — and modulo two it is false, refuted by the inversion centre.

 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. The classification

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.

 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. Lattices

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.

 glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn. Operations

Three reflections, and never four

Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else.

 3.4.6.4 and its dual. The tiling in pale outline with its dual drawn over it: one dual vertex at the centre of every tile, one dual edge across every shared edge, and one dual tile round every vertex. 3.4.6.4 has 3 kinds of tile and one kind of vertex; its dual has one kind of tile and 3 kinds of vertex, and the congruence of those tiles is checked rather than eyeballed — every dual face presents the same cyclic sequence of squared edge lengths, compared exactly. That swap is what the eleven duals are for: read one way the list classifies tilings with all vertices alike, read the other it classifies tilings with all tiles alike. The classification

Eleven duals, one tile each

Swap the vertices of a uniform tiling for its tiles and the eleven come back as eleven tilings by a single repeated shape. Three of those shapes are pentagons — which is worth pausing over on a site whose other essays prove that five-fold symmetry cannot exist.

 P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn. Operations

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

 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. Lattices

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.

 hexagonal: 0.5 and 0.577. The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid. Lattices

Covering and packing want different lattices

A lattice has two natural radii — the largest spheres on its points that do not overlap, and the smallest that leave no gap — and both are radii of the same Voronoi cell. In the plane one lattice is best at both. In space the best packer and the best coverer are different lattices, and they are duals of one another.

 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. What a lattice forbids

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.

 p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers. The classification

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

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