Crystallography & Symmetry
patterns derived from their groups
Concept

Binary quadratic form — where it appears

A binary quadratic form is an expression a x² + b x y + c y² in two variables, which is exactly the metric of a plane lattice written in one basis. Changing the basis changes the three numbers and leaves the lattice alone, so a lattice is a class of forms.

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

Also named here as discriminant — 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 Discriminant −23 at the prime 2: the crater and three levels down. The lattices of discriminant minus 23 and of 4, 16 and 64 times that, joined by index-two sublattices. Three lattices form the crater, a cycle in which each has two neighbours of its own discriminant. Each crater lattice has one sublattice going down, and every lattice below has one edge up and two down. The levels hold 3, 3, 6, 12 lattices.

One way up and p ways down

Every plane lattice has exactly p + 1 sublattices of prime index p. Restrict an integer form to each and the discriminant can do only three things, so the lattices become the vertices of a graph with a definite shape: a ring of lattices at the top, and below it a tree in which every lattice has one way up and p ways down. Whether the ring exists at all is decided by how p factors, which is why the square lattice has two copies of itself at index five and none at index three.

lattices · Moduli

Named alongside it

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

Class numberDiscriminantEquivalence classHexagonal latticeLattice reductionModular regionModuli spaceNormal formSimilaritySublatticeTheta series

All concepts