Crystallography & Symmetry
patterns derived from their groups
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The thread: 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.
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.

 -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.

 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.

 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.

 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.

 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.

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