Every essay
Operations
Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.
What a symmetry actually is
Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.
The four motions of the plane
Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.
Why it is a group and not a list
The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.
The orbit is the pattern
A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.
Lattices
The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.
The lattice underneath
Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.
Five lattices, and no others
A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long.
The cell is a choice, the lattice is not
Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.
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.
The seventeen
Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.
Reading Hermann–Mauguin
p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.
p3m1 and p31m
Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence.
The motif must be a comma
A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption.
Seven friezes
The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.
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.
The crystallographic restriction
A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.
Why five-fold is impossible
A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.
Order without repetition
Penrose tilings and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.
Penrose tilings
Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.
Inflation, and where the golden ratio comes from
A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio.
Order is not periodicity
For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.
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.
The reciprocal lattice
Nobody has seen a space group. Crystals are read from where they scatter, and where they scatter is a second lattice in which long has become short and short has become long.
Systematic absences
The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.