Crystallography & Symmetry
patterns derived from their groups
Series

Moduli — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

    The space every lattice lives in

    Five lattices in the plane is the number of *kinds*. The number of lattices is a continuum — and it has a shape: one two-dimensional region with two corners, three edges and an interior, where the five kinds turn out to be a region, three arcs and two points rather than five things of one sort.

    part 1 · lattices
  2. The region, and its copies. Words in S and T up to length 4, each carrying the region somewhere else. The copies do not overlap and they do not leave gaps: the upper half-plane is tiled by them, one copy per change of basis. That is the whole content of the claim that reduction picks a canonical basis — every basis of every lattice is in exactly one copy, and reduction is the walk back to the shaded one.

    Two moves reach every basis

    A lattice has infinitely many bases and reduction picks one. Why it can is a fact about a group with two generators and two relations — and the fundamental region tiles the plane with its own copies, one per basis, which is what makes the walk home finite.

    part 2 · lattices
  3. A lattice placed in the region, and its distance to each special shape. The modular region, with the two special points marked — the square lattice at the top of the arc and the hexagonal one at its corner — and a third lattice placed by reducing its form. The distances are hyperbolic rather than Euclidean, and the choice is forced rather than aesthetic: a distance between lattice shapes has to be unchanged by every change of basis, and the hyperbolic metric is the one defined by being invariant under exactly that group. Writing the same lattice down on three other bases and measuring again gives the same two numbers to the last digit.

    How far one lattice is from another

    A crystal that is nearly hexagonal twins where an exactly hexagonal one would not, and 'nearly' does real work in that sentence. Giving it a number needs a distance that no change of basis can move — which forces the geometry to be hyperbolic rather than flat.

    part 3 · lattices
  4. 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.

    part 4 · lattices
  5. 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.

    part 5 · lattices
  6. 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.

    part 6 · lattices
  7. 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.

    part 7 · lattices

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