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.

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.

Two 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.

Where the exactness stops

Three places, and they fail differently.

The plane, not space. Everything computed here is two-dimensional: seventeen wallpaper groups rather than two hundred and thirty space groups. The three-dimensional case is the same argument with more bookkeeping, and where an essay discusses it, it says so rather than pretending the code went there.

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.