Crystallography & Symmetry
patterns derived from their groups
Concept

Hermite normal form — where it appears

The triangular form an integer matrix reduces to by row operations, which names the sublattice a set of vectors generates. It is the right way to enumerate sublattices and a poor basis to work in, since the operations of the parent look unrecognisable in it.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Sublattices of index n in the plane. For each index up to 12: the number of sublattices found by building every Hermite normal form of that determinant, and the number the Dirichlet series ζ(s)ζ(s−1) predicts — the sum of the divisors in the plane, and a longer sum in space. The two columns are computed by routines that share no code, and the figure does not appear at all if any row disagrees.

How many ways there are to thin a lattice

A sublattice of index n keeps one lattice point in n, and there is never only one way to do it. In the plane the number of them is the sum of the divisors of n; in space it is a longer sum; and both are counted here by writing every one of them down.

lattices · Sublattices Every plane group from at most 4 operations. For each group, the fewest operations that generate the whole of it — the point operations and both lattice translations, since a group that does not reach its own translations is a different group. The floor is the abelianisation's number of invariant factors, which no group can beat, and the search is exhaustive over the operations within one cell of the origin. 14 of the seventeen meet their floor, which settles those exactly; the other 3 need more than the abelian argument can see, and p3m1 needs three where its abelianisation is cyclic.

How few operations make a pattern

A plane group is infinite, and a handful of its operations is enough to rebuild all of it. How small a handful is a question with a floor from the abelianisation and a ceiling from an exhaustive search, and for fourteen of the seventeen the two numbers meet.

operations · Presentations p = 2: 1, 3, 6, 12, 24 vertices at each distance. Every sublattice of index a power of 2, up to scale, joined when one contains the other with index 2. From the whole lattice there are 3 ways down, because a sublattice of index 2 is a line over the field of 2 elements and there are 3 of those; from each of those there are 3 again, one of which is the way back. So the counts are 1, 3, 6, 12, 24 — that is (2 + 1)·2^(k−1) — and the graph has no cycles, both of which are checked on every vertex whose whole neighbourhood was grown rather than read off the picture. The object is the Bruhat–Tits tree of the p-adic plane, and it is what the set of sublattices is rather than how many there are.

Every way down, and no way round

There are as many sublattices of a given index as the index has divisors, and counting them is where that essay stopped. This one asks what they are to each other, and the answer is a shape: an infinite tree in which every vertex has exactly p + 1 neighbours and no path ever comes back.

lattices · Sublattices At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

The same group in a bigger cell

A subgroup usually gives something up. An isomorphic subgroup gives up nothing but scale — the same plane group again, on a coarser lattice — and the indices at which that is possible turn out to be the values of a quadratic form.

space-groups · Isomorphic subgroups Sublattices of index n, in space. How many sublattices a three-dimensional lattice has at each index, beside the plane's answer, with the Hermite enumeration and the coefficient of ζ(s)ζ(s−1)ζ(s−2) in separate columns. The two are computed by routines sharing no code, and a row where they disagreed would be a failure rather than a result. The last column counts the ones that survive every operation of the cubic group, and it is almost always empty.

The three that stay cubic

A lattice in space has far more sublattices than one in the plane — 651 of index sixteen against 31 — and almost none of them keeps the symmetry it came from. The ones that do exist at indices m³, twice m³ and four times m³, there is exactly one at each, and they are the primitive, face-centred and body-centred cubic lattices, arrived at by asking which sublattices keep a symmetry rather than by enumerating centrings.

lattices · Sublattices 342 unlabelled spots, cell volume 52. A bag of 342 reflection positions with no indices on them, collected out to a bound of 3 on each index. Their pairwise differences generate the reciprocal lattice; a basis of that is taken by integer elimination and then reduced, and the reduced basis is printed. Its determinant is 52, which is the volume of the cell the reflections were computed from — so the cell has been recovered from positions alone, with no intensity used anywhere.

A cell from a bag of spots

A single-crystal experiment returns a list of directions with no labels on them. Recovering the cell is recovering the lattice those directions generate, and the whole of it is take differences, reduce, read the answer. What no quantity of data settles is whether the lattice found is the true one or a sublattice of it.

applied · Indexing Σ5: three lattices in one picture. Two copies of the square lattice turned by 36.87 degrees against one another — one drawn pale, one drawn in the second colour — with the points they share ringed. The fine dots are the lattice generated by both together, the DSC lattice, which contains each crystal with index 5 exactly as the coincidences sit inside each crystal with index 5. Three lattices nested at the same index, and the middle one is the crystal.

The dislocations a boundary allows

Two crystals meeting at a coincidence angle share one lattice and generate another. The second is where a boundary's own defects live, its shortest vector is one over the square root of the index, and a dislocation's energy is the square of that.

applied · Interfaces

Named alongside it

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

SublatticeIndexCentringDivisor sumSubgroupSuperstructureAbelianisationBasis reductionBravais latticeBruhat tits treeBurgers vectorClosure

All concepts