Every crystal fits one of just 14 repeating point patterns
Salt, diamond and quartz look nothing alike, yet the scaffolding beneath every crystal belongs to one of only 14 point arrangements in three dimensions. Auguste Bravais catalogued them in 1850. Flatten space to a plane and only five survive; add a fourth dimension and the count jumps to 64.
A Bravais lattice is an endless grid of points built by repeating a few fixed shifts, so that the view from any point is identical to the view from every other. In three dimensions, three basic step vectors, pointing in different directions but not necessarily at right angles, generate the whole array when added together in whole-number amounts. Several different sets of steps can produce the same lattice.
Crystallographers use the lattice as a frame. At each point sits a basis, also called a motif: an atom, a molecule or even a strand of polymer. The lattice says where copies go; the motif says what is copied. Lattices counted as equivalent when their symmetry groups match yield five types on a flat plane and 14 in space, and those 14 symmetry groups form a small subset of the 230 space groups.
The repeating tile is the unit cell, a parallelogram in two dimensions and a slanted box in three. The primitive cell is the smallest possible tile and holds exactly one lattice point, but it can hide the pattern's symmetry. A conventional cell trades size for clarity: it shows the full symmetry and contains one, two, three or four times as many points. Those extra points define the centering types. Primitive cells have points only at the corners; body-centred ones add a point in the middle; face-centred ones add one on every face; base-centred ones add points on a single pair of opposite faces.
Pairing the seven three-dimensional lattice systems with these centering options gives more combinations than are really distinct. A monoclinic body-centred lattice, for instance, is just a monoclinic base-centred one viewed along different axes. Removing such duplicates leaves the famous 14. Four-dimensional space holds 64 lattices spread over 23 crystal families, 23 of them primitive and 41 centred.
Source: Bravais lattice
