About

This is a growing collection of illustrated essays about symmetry — in wallpaper, in friezes, in crystals, and in the quasicrystals that were supposed to be impossible. Each takes a single idea and draws it until the argument is visible, and every pattern in every picture was generated from a group and then checked back before it was allowed to appear.

The round trip

This subject has a property that almost none of its neighbours have: its central questions are decidable. A pattern either has a symmetry or it does not, and finding out is integer arithmetic. There is no tolerance to choose and no residual to interpret. So the verification here can be stronger than a check — it can be a round trip.

1. Generate. Take a small motif and apply a group's operations to it until applying another produces nothing new. The result is the pattern.
2. Forget the group. Hand the bare point set to a detector that examines every candidate operation the lattice permits and keeps the ones that map the set to itself.
3. Compare, in both directions. The detected set must equal the generating set exactly.

Both directions earn their place, and the second is the interesting one. Detecting fewer symmetries than were generated means the motif was not really invariant — that tends to look wrong, so a reader would catch it. Detecting more means the motif was accidentally too symmetric, and the picture is illustrating a different group from the one named underneath it. Nothing about the picture betrays that. It is the characteristic error of hand-drawn pattern figures and it is invisible to the eye, which is the entire argument for computing it instead.

Where being decidable stops

That property is not universal in the subject, and knowing where it fails is part of knowing what it is worth. Whether a given pattern has a given symmetry is decided by arithmetic; whether a finite set of tiles covers the plane at all is decided by no procedure whatever, and cannot be. Between those two sit the questions this collection answers by counting rather than by deciding — how many arrangements a local rule permits, how much of a lattice's own symmetry a defect can carry — and the questions it answers by measurement, where a search on a grid has to be refined until the answer stops moving.

Each of those is stated where it arises, with what was computed and what was quoted kept apart. The short version: a group is decidable, a pattern is decidable, a tiling is not, and a count is neither.

One step in from that boundary sits a question the collection keeps returning to: what can a condition imposed on a bounded piece of a pattern force about the whole of it? Sometimes everything — a test run on one tile's boundary settles a tiling of the entire plane, and a walk round a defect returns an invariant of the whole crystal. Sometimes almost everything and then nothing: a shape can be surrounded twice over and tile nothing, and a tiling by congruent tiles can still need two kinds of tile. And sometimes the local fact is the only thing a measurement ever sees, and the global one has to be levered out of it — which is what a phase-determination procedure is.

Why the motif is a comma

The nineteenth-century ornament plates draw their motif as a comma rather than a dot, and the reason turns out to be computable. A single dot verifies only eleven of the seventeen groups. Its orbit under p3 acquires three mirrors and a pair of glides, so the picture is p31m; under p3m1 it acquires a six-fold centre and is really p6m; and under p1 it acquires an inversion centre, because the midpoint between any dot and its own lattice translate is one — so a lone dot can never illustrate the group with no symmetry at all.

Two points fix everything but p1, for the same reason about midpoints. Three points in no particular arrangement fix all seventeen. The motif used throughout this site is three points, and it is a comma for a reason that was rediscovered rather than inherited.

Three routes, not one

A pattern figure asserts its group directly from the point set. A diffraction figure computes what that same point set would scatter, by the sum a crystallographer writes down, and the systematic absences fall out — alternate reflections along a row cancelling exactly where a glide plane is present. Two calculations that share nothing but the atom positions, arriving at the same symmetry.

That matters because it is how the subject is actually done. Nobody has seen a space group. They are inferred from where a crystal scatters and, just as informatively, from where it does not.

The third route throws the geometry away entirely. Keep which atoms are joined and forget where they are, and what is left is a periodic graph — a few vertices, a few edges and a pair of integers on each — with no lengths and no angles anywhere in it. Put every vertex at the average of its neighbours, which is one linear system per coordinate and exact in the lattice basis, and the group comes back: the same detector, run on coordinates that were computed from incidences rather than measured. The essays on nets do this, and it is the least expected of the three, because a group of motions of a plane is exactly the kind of thing that ought to need a plane.

Where the exactness stops

Three places, and they fail differently.

The round trip is two-dimensional. The detector that rediscovers a group from a point set works on plane lattices: seventeen wallpaper groups rather than two hundred and thirty space groups. Two of the essays do carry exact arithmetic into three and four dimensions — the point symmetries of a cubic lattice, enumerated as signed permutations, and the integer matrix of order five that four dimensions permits — but that is a separate and much smaller piece of machinery, and both essays say which of the two produced each number.

Aperiodic patterns are not decided. A Penrose tiling has no lattice, so the integer machinery does not apply to it. The aperiodic figures are constructed by inflation and their properties are measured — tile ratios, vertex types, diffraction peaks — rather than proved. Where a number in an essay came from measurement it is described as a measurement.

A drawing is a drawing. The check proves that the point set has the symmetry claimed. It cannot prove that the point set is the one a reader sees, and a label placed carelessly can still mislead. That failure mode is caught by looking, which is why every figure on this site also gets looked at.

Vector, not raster

Every figure is generated SVG, recolours for the dark theme, and can be checked by the layout gates. Nothing here is a bitmap of a pattern, which would be both heavier and unexaminable.

On being wrong

Corrections are welcome and will be made. An elegant wrong answer is worse than none in this subject, because a pattern with the wrong caption looks exactly like a pattern with the right one.