Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Symmetry is decidable — page 6

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
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.

 p3m1 and p31m, told apart without a picture. The two groups this site returns to most often: same point group, same lattice, same number of operations, and distinguished in every other essay here by where their mirrors sit relative to the lattice — which is a fact about the plane. Abelianised, they are ℤ2 and ℤ6, which are not isomorphic. That difference is a fact about the groups: no change of basis, no redrawing and no relabelling can carry one to the other, and the argument never mentions a mirror line. Operations

What is left when the order is forgotten

Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate.

 Every plane group from at most 4 operations. For each group, the fewest operations that generate the whole of it — the point operations and both lattice translations, since a group that does not reach its own translations is a different group. The floor is the abelianisation's number of invariant factors, which no group can beat, and the search is exhaustive over the operations within one cell of the origin. 14 of the seventeen meet their floor, which settles those exactly; the other 3 need more than the abelian argument can see, and p3m1 needs three where its abelianisation is cyclic. Operations

How few operations make a pattern

A plane group is infinite, and a handful of its operations is enough to rebuild all of it. How small a handful is a question with a floor from the abelianisation and a ceiling from an exhaustive search, and for fourteen of the seventeen the two numbers meet.

 Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear. Lattices

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

 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.

 Every group decided by a window of radius 1. For each of the seventeen, the radius at which a round window on the pattern admits exactly the group's own operations and no others — with the numbers it admits at each smaller radius beside it. Two opposite failures are visible. Most groups under-report at small radii, because an operation carrying points out of the window cannot be tested at all; cm over-reports, admitting operations the pattern does not have. The groups that take longest to settle are the ones distinguished by a glide, which moves a point half a cell before anything can be compared. The classification

How much pattern is enough

Every claim here about a pattern's group is a claim about an infinite pattern. A reader sees a patch. Measuring what a finite window can decide gives a number — about one cell's radius — and two opposite ways of being wrong on the way there.

 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.

 Why the seventeen is a number at all. The classification is finite because three counts in a row are finite, and the first two are where the work is. Finitely many lattice types, because a lattice's symmetry group is a finite group of integer matrices; finitely many such groups, by Minkowski's lemma and his bound; and finitely many ways to attach translations to each, which is the extension problem. Every step is a count this site makes elsewhere — five, thirteen, seventeen — and this is the reason each of those searches was allowed to stop. What a lattice forbids

Why there is a list at all

Five lattices, seventeen groups, thirty-two classes, two hundred and thirty. Every one of those counts came out of a search that had to know when to stop, and the reason it could stop is a divisibility Minkowski proved in 1887.

 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.

 Every way regular polygons can fill a turn. The seventeen multisets of regular polygons whose interior angles add to exactly 360°, listed with the sum that qualifies each of them. They are found by a search over sizes from three upward: the largest polygon that can appear is the forty-two-gon, which needs a triangle and a heptagon beside it, and the search stops there because the smallest interior angle is a third of a turn so at most six polygons can meet. Nothing here is a table looked up — the list is the output of the search, and every count on the page downstream of it is counted from this one. The classification

Twenty-one vertices, eleven tilings

Regular polygons meeting at a point must fill exactly a turn, which is a Diophantine equation with seventeen answers and twenty-one cyclic arrangements. Ten of the twenty-one tile nothing at all — and the argument that kills them counts places round a polygon rather than measuring anything.

 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.

 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 Fibonacci chain: p(n) = n + 1. The number of distinct windows of each length in the Fibonacci chain, measured by sliding a window along 46,368 tiles. Every count is checked against the same count on half the chain, and only lengths where the two agree are drawn — a factor count on a finite word is otherwise a lower bound wearing the clothes of an answer. Order without repetition

n plus one, and no fewer

Slide a window along a chain and count what it can show. A periodic chain runs out of new views; an aperiodic one never does; and the fewest an aperiodic chain can manage is one more than the window's length — which is exactly what the Fibonacci chain manages.

 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.

 P222: 16 descriptions of one structure. P222 has 4 operations in a cell. 8 origins leave every one of them exactly where it was, and 8 linear parts of the lattice's holohedry normalise the group, so its Euclidean normaliser has 64 elements per cell and the index is 16. That index is the number of coordinate lists that describe one and the same arrangement of atoms. Each was applied to a motif and the resulting point sets compared: the numbers differ and the sets are identical, which is the check that makes the count mean anything. Operations

One crystal, and sixteen coordinate lists

Two structure reports can disagree in every number and describe the same arrangement of atoms, because a space group does not fix its own origin or its own axes. How many genuinely different lists there are is the index of the group in its Euclidean normaliser — a number, computable, and the thing a structural database has to divide out before it can say two entries are the same compound.

 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.

 incommensurate: "dense on a line". The shortest non-zero vector a subgroup contains, as the search widens, against the square lattice drawn flat behind it as a control. For a lattice the answer is constant: the shortest vector is the shortest vector, and looking further finds nothing nearer. For a subgroup that is not a lattice it falls without limit, because the convergents of a continued fraction give integers making the combination arbitrarily small. This one falls from 0.414 to 1.2e-2 over bounds 1 to 64, which is the verdict "dense on a line" arrived at by measurement rather than by reading a definition. Nothing here is decided by asking whether a ratio is rational; the ratio is a float and the question would be undecidable of one. Lattices

Discrete, or dense, and nothing between

Every count in this collection rests on a hypothesis nobody states, because it is built into the word lattice: the translations of a pattern form a discrete subgroup of the plane. Drop it and the counts do not become larger — they stop existing, because the object stops being a lattice. A subgroup of the plane is one of five things, and only two of them are lattices.

 One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are. Order without repetition

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

 Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092. How it is known

How many reflections there are

The reciprocal lattice is infinite and a measurement is not. A wavelength cuts a sphere out of it, the number of points inside is the sphere's volume times the cell's, and symmetry then removes all but a fraction — though never quite the fraction a division would give.

 a general quadrilateral tiles. A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested. The classification

Which shapes tile by themselves

Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.

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