Exactly seventeen, and not one more.
There are exactly seventeen ways to repeat a pattern across a flat surface. Not seventeen that anybody has found so far — seventeen that can exist, with a finite argument and no eighteenth case. That is an unusual kind of sentence for a subject about ornament, and these essays are about where sentences like it come from: the lattices that constrain what a symmetry can be, the rotations they forbid, and the crystals that turned out to have one anyway. And then about what a group decides once it is there — which functions it leaves alone, which quantities a crystal may have at all, and what it is allowed to do when it stops being symmetric.
Newest
what has arrived most recently · everything, by arrival
The relation that is an equality · The relation that can say no · The denominator a group actually needs · The screw a dimension does not have · Two patterns laid over one another · The axis a product lies on
The nine fields
the order the argument builds · all of them, with their ladders
41 essays
Operations
Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.
43 essays
Lattices
The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.
42 essays
The classification
Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.
39 essays
What a lattice forbids
Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.
40 essays
Order without repetition
Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.
42 essays
How it is known
A crystal's structure is read from the pattern it scatters. The reciprocal lattice, systematic absences, and what an experiment can and cannot see.
41 essays
Into space
Seventeen becomes two hundred and thirty. Screw axes, glide planes, and the translations that no choice of origin can remove — the operations a flat surface has no room for.
42 essays
What symmetry decides
Thirty-two classes, enumerated rather than listed — and then the one thing they are for outside crystallography's own bookkeeping: which components of a physical property a crystal is permitted to have before anybody measures it.
63 essays
Symmetry at work
Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.
Start anywhere
one ladder per idea · all 69 · all 393 essays
Operations — Fundamental domain · What symmetry is · Composition · Presentations · Normalisers · Subgroups · Counting
Lattices — Lattice · Wigner–Seitz cells · Sublattices · Moduli · The fourteen Bravais lattices · Lengths · Centring
The classification — Seventeen · Tilings · Decidability · Flat space · Subperiodic · Cohomology · Colour · Friezes · Ornament · Isohedral
What a lattice forbids — Curvature · Finiteness · Local symmetry · Restriction · Finite groups
Order without repetition — Aperiodic · Quasicrystals · Complexity · Entropy · Modulation · Monotile
How it is known — Diffraction · Statistics · Direct methods · Homometry · Resolution · The Patterson function · Accidental symmetry · Disorder · Friedel's law
Into space — Isomorphic subgroups · K symmetry · Space groups · Screws and glides · Chirality · Settings · Landau
What symmetry decides — Crystal classes · Representations · Invariants · Neumann's principle · Properties · Curie · Magnetic · Optical
Symmetry at work — Domains · Nets · Packing · Interfaces · Rigidity · Forms · Twinning · Defects · Growth · Indexing
Themes running through
themes, not chapters
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.
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.
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.
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.
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.
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.