Crystallography & Symmetry
patterns derived from their groups
Generator

Sublattices of index n in the plane

Sublattices of index n in the plane
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.

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.

10 essays call sublattice-count. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Lattices

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.

 Which indices have a square sublattice. For each index up to 26: how many sublattices of the square lattice are themselves square, found by testing whether the quarter-turn maps each one onto itself; the same count as a sum over divisors, +1 for each divisor one more than a multiple of four and −1 for each one less; and the ways of writing the index as a sum of two squares. The three agree at every row, which is Fermat's theorem — and it says that 3, 7 and 11 have no square sublattice at all while 5, 13 and 17 have two. Lattices

The sublattices that stay square

A sublattice of the square lattice is itself square exactly when its index is a sum of two squares — so index five has two and index seven has none, and which superstructures a surface can form is decided by a theorem of Fermat's about primes.

 What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order. Operations

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

 The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry. Lattices

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

 What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order. Operations

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

 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. Lattices

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.

 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. Into space

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.

 Sums of two squares, arriving as superstructures. Which indices admit a sublattice of the same shape as the square lattice, drawn as a bar per index whose height is how many there are. The pattern is not a pattern about lattices at all: an index works exactly when it is a sum of two squares, because a similar sublattice of the square lattice is multiplication by a Gaussian integer and its index is that integer's norm. The indices that work up to 30 are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, and the same list is produced here a second time by factorising rather than by searching, with the two required to agree. Lattices

The sublattices that are the same shape

Thinning a lattice usually changes its shape. Sometimes it does not: the sublattice is the parent rotated and scaled, and a drawing of it alone would be a drawing of the parent. Which indices allow it turns out to be a question Fermat answered in 1640.

 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. Lattices

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.

 How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have. Lattices

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

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