Crystallography & Symmetry
patterns derived from their groups
Concept

Dirichlet series — where it appears

A way of holding a whole sequence at once: the terms are divided by the index raised to a variable power and added. Its use is that a sequence whose values at coprime indices multiply gives a series that factors into one term per prime, so a row of counts becomes a product.

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

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 A row of coefficients for every group. For eight of the seventeen, the number of sublattices of each index that the group's point group carries to itself. A nought means the group has no copy of itself at that index at all. p1 and p2 preserve every sublattice, so their rows are the counts of sublattices themselves — 1, 3, 4, 7, 6, 12 — which is the sum of the divisors. The rows thin out as the point group grows, and p4m and p6m have almost nothing in them. Every row is a sequence a Dirichlet series can be built on, and the next figure is what that series factors into.

A row written as a product

Every group's copies of itself sit at a row of indices, and every row so far has been read one entry at a time. Counting all of them at once turns a row into a Dirichlet series, and every one of the seventeen rows factors into a product over the primes — which is the statement that a copy is a chain of maximal steps, written as arithmetic. The plainest group of all has the most famous series in mathematics.

space-groups · Isomorphic subgroups

Named alongside it

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

IndexSublatticeBravais latticeCentringCountingDivisor sumGaussian integerHermite normal formHolohedryIsomorphic subgroupMaximal subgroupMultiplicative function

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