Crystallography & Symmetry
patterns derived from their groups
Field

Lattices

The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.
hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell

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.

 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

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.

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

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.

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