Theme

The theme: Symmetry is decidable — page 14

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.

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.

Into space

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.

One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against.

The classification

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

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