Themes
A field is a partition: every essay is in exactly one, and the fields between them account for the whole collection once. A theme is the opposite kind of object. An essay carries as many as apply, a theme carries as many essays as happen to share the motif, and the interesting ones are the ones that show up in fields with no business being related.
So the fact worth putting on the card is not the count but the spread: which fields a theme reaches. A motif tagged in one field is a subject; a motif tagged in seven is the thing this collection is actually claiming. The tags under each theme below are computed from the essays carrying it — nothing here declares which fields a theme belongs to, and a theme that stops crossing a field stops saying so on its own.
The sizes are deliberately uneven and the extremes are the point. This site's plan holds itself to no theme above about a fifth of the collection or below eight essays at size, which the largest and smallest below straddle: a motif on nearly every essay is close to being a description of the site rather than a thread through it.
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'.
Operations 25Lattices 36The classification 39What a lattice forbids 31Order without repetition 23How it is known 28Into space 26What symmetry decides 36Symmetry at work 46
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.
Operations 3Lattices 10The classification 7What a lattice forbids 25Order without repetition 9How it is known 2Into space 3What symmetry decides 4Symmetry at work 10
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.
Operations 21Lattices 14The classification 20What a lattice forbids 6Order without repetition 13How it is known 2Into space 7What symmetry decides 7Symmetry at work 18
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.
Operations 3The classification 5What a lattice forbids 1Order without repetition 1How it is known 1Into space 4
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.
Lattices 1The classification 3What a lattice forbids 2Order without repetition 35How it is known 1Symmetry at work 2
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.
Lattices 7The classification 2What a lattice forbids 2Order without repetition 10How it is known 41Into space 2What symmetry decides 2Symmetry at work 8
No origin removes it
An operation's translation splits in two: a part that belongs to the operation, and a part that only records where somebody put the origin. Almost every argument about space groups is about telling them apart, and the first half of the split is the whole difference between a rotation and a screw.
Operations 10Lattices 1The classification 3How it is known 7Into space 21What symmetry decides 1Symmetry at work 2
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.
Operations 37Lattices 38The classification 32What a lattice forbids 30Order without repetition 15How it is known 32Into space 34What symmetry decides 39Symmetry at work 57
The graph remembers
Throw away where the atoms are and keep only which of them are joined. The lengths go, the angles go, and the group survives the loss — recoverable from the incidences alone, by putting every vertex at the average of its neighbours.
Operations 3The classification 2What a lattice forbids 2Into space 1Symmetry at work 13
The same arithmetic, renamed
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.
Operations 15Lattices 20The classification 10What a lattice forbids 17Order without repetition 11How it is known 10Into space 15What symmetry decides 12Symmetry at work 26
The group acts on functions
A rotation carries an atom onto an atom, and it carries a density, a displacement or a wave onto another one. The second action is linear, so the group becomes a set of matrices, and questions that look analytic — which levels must coincide, how many independent components a property has, how a shell of neighbours splits — become counts of whole numbers.
Operations 4The classification 1What a lattice forbids 1Order without repetition 1Into space 10What symmetry decides 16Symmetry at work 6