Crystallography & Symmetry
patterns derived from their groups
Concept

Class number — where it appears

How many inequivalent lattices share a given metric determinant, counted up to a change of basis by an integer matrix of determinant one. It is finite, it is computed by enumerating reduced forms, and it is finer than anything a congruence can measure — which is what the genus counts instead.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Three lattices no congruence can separate. The three reduced forms of discriminant minus twenty-three, with the integers each represents. The principal form represents one and the others do not; the others represent two and it does not. So they are genuinely different lattices — and they represent exactly the same residues modulo twenty-three, so they are in one genus and no congruence condition of any kind distinguishes them.

Lattices that agree at every prime

Counting the plane lattices with a given metric determinant is a class number. Above it sits a coarser count — the genus, which is what congruences can see — and for most small determinants the two agree. At discriminant minus twenty-three they part: three lattices representing exactly the same residues modulo everything, and different integers. No argument modulo any number can tell them apart, and they are not the same lattice.

lattices · Moduli The turns a rolled sheet can make are square roots. Every primitive rolling of a square and of a hexagonal sheet with at most 66 cells in a repeat, one mark per distinct turn. Across is the number of cells n₀ in one repeat of the tube; up is the turn, as a fraction of a circle, of the screw that climbs one n₀-th of the repeat. Most columns are empty, and the occupied ones hold two or four marks rather than anything up to n₀: on the square sheet the turns are exactly the square roots of minus one modulo n₀, on the hexagonal sheet the roots of the corresponding congruence for minus three. The pattern is symmetric about half a turn because a root and its negative are the two hands of one tube.

A rolled sheet turns by a square root

Roll a square sheet into a tube of thirteen cells and its screw could, on paper, turn by any of twelve fractions of a circle. It turns by five thirteenths or eight, and nothing else, because five squared plus one is a multiple of thirteen. Every rolled tube's screw is a square root of minus the sheet's determinant — which is why handed tubes appear only at the orders Fermat allows, why they always come in both hands, and why the crystallographic ones need an accident.

restriction · Finite groups 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.

Binary quadratic formDiscriminantGram matrixQuadratic formTheta seriesCrystallographic restrictionDeterminantEnantiomorphEquivalence classHelixHexagonal latticeLattice

All concepts