Crystallography & Symmetry
patterns derived from their groups
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The thread: 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'.
-2-10121-fold2.000integer — allowed2-fold-2.000integer — allowed3-fold-1.000integer — allowed4-fold0.000integer — allowed5-fold0.618not an integer6-fold1.000integer — allowed7-fold1.247not an integer8-fold1.414not an integer9-fold1.532not an integer10-fold1.618not an integer11-fold1.683not an integer12-fold1.732not an integerrotationtrace2 cos(2π/n) — a whole number only five timescomputed, not tabulated1, 2, 3, 4, 6 What a lattice forbids

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.

 p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified The classification

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.

 obliqueno constraint on lengths or anglerectangularangle 90°, lengths freecentred rectangularequal lengths, angle freesquareequal lengths, angle 90°hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell Lattices

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.

 rotate both ways and add|shortest| × 0.618shorter than the shortestso no such lattice existsthe descent argument, drawn5-fold What a lattice forbids

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.

 p4msquare lattice · 8 operations per cellelements marked The classification

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.

 115 present6 absentthe glide's signaturecomputed from the atom positions, not from the grouppg How it is known

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.

 one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tilesthe substitution rule applied to a single tile Order without repetition

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.

 first: mirrorthen: mirrorlands on: rotation of order 4closure is what makes it a group, not a listp4m Operations

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.

 p3m1hexagonal lattice · 6 operations per cellelements marked The classification

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.

 p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth The classification

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.

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