Crystallography & Symmetry
patterns derived from their groups
Concept

Reduced cell — where it appears

A cell chosen by a rule that depends on the lattice alone, so that two crystallographers with different bases still agree. Reduction is what turns the question of whether two published cells describe one crystal into a comparison of six whole numbers.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Two cells of equal area on one rhombic lattice. One lattice — the rhombic lattice that cm sits on — with two parallelograms drawn on it: the conventional cell, and a sheared cell whose edges are the integer combinations (1, 0) and (1, 1) of it. The points are identical in both outlines; only the description changes. Each cell's contents were counted by writing every lattice point in that cell's own coordinates and sharing each one out between the cells that meet at it — a quarter at a corner, a half on an edge, one inside — and the totals come to 1 and 1, which are the determinants of the two matrices. The alternative has determinant one, so its inverse is integral and it generates exactly the same lattice; that is the whole condition, and it is why a lattice has infinitely many bases and no arithmetic can prefer one.

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.

lattices · Lattice Reducing a basis. An awkward basis and the reduced one Gauss's algorithm returns. Both describe the same lattice — the change of basis has determinant one — and the reduced pair is the shortest vector together with the shortest independent of it, checked against an exhaustive search.

Reduction, and the shortest basis

Every lattice has infinitely many bases and no arithmetic picks a preferred one — until a rule is imposed. Reduction is that rule, it terminates in a handful of steps, and it is what lets a database decide whether two reported crystals are the same crystal.

lattices · Lattice Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it.

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

lattices · Lattice (17, 5) and (23, 7) reduced in 3 steps. Lagrange's reduction, run on the basis (17, 5), (23, 7). Each step subtracts a whole multiple of the shorter vector from the longer and swaps them; after 3 steps neither can be shortened by the other and the pair is reduced. The faint arrows are the intermediate bases and the solid pair is the answer, of length 1.41. The procedure always terminates and always finds the shortest vector, and in the plane that is a theorem rather than a hope.

The shortest vector, and where it stops being easy

Two moves find the shortest vector of a plane lattice, and they always terminate. Nothing on this site has ever needed more, because every lattice here has two or three dimensions. In general the same question is NP-hard, the best polynomial procedure returns an answer that may be exponentially too long, and an entire branch of cryptography is built on the gap.

lattices · Lattice Four vectors summing to zero, and six numbers on the edges. A superbasis is the three basis vectors together with their negated sum, so the four sum to nothing and their pairwise products sit on the six edges of a tetrahedron. Selling's rule is: while any edge is positive, apply one transformation. The right panel is the same lattice reduced, with the vanishing parameters marked — and a vanishing parameter is a face the Voronoi cell does not have.

A reduction with one rule

Niggli's reduction is eight numbered conditions with sub-cases, applied in order until none applies. Selling's is a single rule on four vectors that sum to zero: while any of six numbers is positive, do one thing. It terminates sooner, its termination is a quantity that visibly falls, and when it stops the six numbers are the Voronoi cell — the pattern of which ones vanish gives Fedorov's five solids and nothing else.

lattices · Wigner–Seitz cells

Named alongside it

The objects these essays reach for when they reach for this one.

Shortest vectorBasis reductionGram matrixLatticeThe Niggli cellUnimodular matrixBravais latticeCentringGauss reductionHolohedryLattice characterLll

All concepts