Crystallography & Symmetry
patterns derived from their groups
Concept

Hexagonal lattice — where it appears

The plane lattice with six shortest vectors and a sixfold rotation, whose form is x² + xy + y². It is the densest packing, the best covering and the minimiser of every lattice sum tried, which is a coincidence of the plane rather than a principle.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as lattice reduction — the same set of essays touches all of them, so they are one junction rather than several.

How many different lattices share a determinant. One bar per determinant: the number of inequivalent integral lattices whose metric has that determinant, which is the class number of the corresponding discriminant. Area does not decide shape — at determinant 1 and 2 there is one lattice each, and by 11 there are four — and the count does not grow steadily either. Each bar is computed twice: once by enumerating the reduced forms directly, and once by reducing every form in a box and collecting the distinct results, which is a search followed by an algorithm rather than a search over answers. The two agree at every bar.

How many lattices share a determinant

Area does not decide shape. The number of inequivalent lattices whose metric has a given determinant is a class number, computed by enumerating reduced forms — and checked by reducing every form in a box and counting what comes back distinct.

lattices · Moduli Every plane lattice, shaded by Σ|v|^(−4). The region every plane lattice is one point of, with each point shaded by the sum of the inverse powers of the lengths of that lattice's own vectors, at equal cell area — dark where the sum is small. The square lattice is the ringed point on the vertical axis and the hexagonal one is at the corners, which are the same lattice on two bases. The minimum is at the corner, and it is at the corner at every exponent tried. That is not the same statement as the densest packing, which is decided by the shortest vector alone: this sum counts every shell, and there was no reason in advance for the two questions to have the same answer.

The lattice that minimises a sum

Packing discs asks about the shortest vector alone. Summing an inverse power over every vector of a lattice asks about all of them at once, and there was no reason in advance for the two questions to have the same answer. They do — at every exponent, and the measurement says by how much and where it cannot say.

lattices · Moduli

Named alongside it

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

Lattice reductionTheta seriesBinary quadratic formClass numberDiscriminantEpstein zetaEquivalence classLattice energyModuli spaceNormal formTruncation errorUniversal optimality

All concepts