Crystallography & Symmetry
patterns derived from their groups

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.

The seventeen wallpaper groupsEvery way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified
Fig. 1 Every way of repeating a pattern across the plane, one cell of each. None of these was drawn. Each was grown — a small asymmetric motif with its group’s operations applied until nothing new appeared — and then handed to a detector that forgets the group and finds every symmetry the point set has. The two must agree exactly, in both directions: too few means the motif was not invariant, and too many means the picture is quietly of a different group from the name underneath it.

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19 essays

turn about a point, and nothing moves at that pointrotation Operations

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.

7 figures hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell Lattices

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.

6 figures -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.

6 figures 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.

8 figures 890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5 Order without repetition

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.

6 figures the crystal latticewhere it scatterslong in one is short in the otheraxis ratio 1.7 How it is known

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.

6 figures flip and slide — neither motion alone is a symmetryglide Operations

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.

8 figures 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.

7 figures 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.

7 figures 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.

7 figures 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.

6 figures 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.

6 figures 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.

7 figures centred rectangularequal lengths, angle freethe arrows are the basis; the shaded region is one unit cell Lattices

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.

6 figures 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.

7 figures 181 reflections10-fold symmetryforbidden to any latticeinteger combinations of ten star vectorsaperiodic Order without repetition

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.

6 figures 1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4 Operations

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.

7 figures a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4 The classification

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.

6 figures 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.

7 figures

Threads 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'.

10 essays

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.

7 essays

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.

9 essays

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.

2 essays

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.

3 essays

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.

3 essays

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.

15 essays