Themes
Exactly this many
Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
The lattice forbids
Periodicity is a strong constraint. It rules out five-fold rotations, most rotation orders above six, and a great many patterns that look perfectly reasonable until the arithmetic is done.
Generated, not drawn
Every pattern on this site is the orbit of a motif under a group, and the group is then rediscovered from the drawing. A picture that was drawn by hand can have symmetries nobody intended.
The motif must be asymmetric
A dot is too symmetric to illustrate most groups — its orbit acquires operations the group never had. The comma on a nineteenth-century wallpaper plate is there for a computable reason.
Order without repetition
Periodicity and order are not the same thing, and separating them is what quasicrystals forced. A pattern can be perfectly determined and never repeat.
From the diffraction back
Nobody has seen a space group. They are inferred from where a crystal scatters and, just as informatively, from where it does not.
Symmetry is decidable
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.