Crystallography & Symmetry
patterns derived from their groups
Concept

Coboundary — where it appears

The change every translation part undergoes when the origin is moved, which alters a description and not the group. Two assignments of translations to a point group differ by one exactly when they describe the same group, so the coboundaries are what a classification divides out before counting.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

18 extension classes, 17 groups. Each of the thirteen arithmetic classes with the number of ways translations may be attached to it — its cohomology — the shape of that group, and how many distinct plane groups the classes come to once the changes of basis that are mere relabellings are quotiented out. The two columns differ in exactly one row, 2mmp, where four extension classes are three groups because two of them are the same group with the axes swapped. No lattice is drawn anywhere in this computation.

Seventeen, without a picture

Every other count of the plane groups has a plane in it — a pattern generated, a domain folded, an orbifold's curvature spent. The same seventeen come out of pure algebra: attach translations to a point group, keep the assignments that close, throw away the ones that differ only by where the origin was put, and add up over the thirteen arithmetic classes.

classification · Cohomology One lattice, two copies of pm. pm on a lattice doubled across its mirrors. The mirrors of the parent are every vertical line; a copy of pm on the doubled lattice has mirrors every other line, and there are two ways to choose which — the solid set or the dashed set. Both are copies of pm with the same lattice and the same point group, and no translation of the parent carries one onto the other, because the translation that would is exactly the one the doubling removed. So a count of invariant lattices is not a count of subgroups, and the gap is visible in the smallest case there is.

A lattice is not a subgroup

Every count of copies so far has counted lattices, and the International Tables count subgroups. One invariant lattice can carry several copies of a group that nothing in the parent carries onto one another — pm's doubled lattice carries two, with its mirrors on the even lines or the odd ones — and how many is a cohomology computation, a first where the classification of the seventeen used a second.

space-groups · Isomorphic subgroups 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 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.

classification · Cohomology Tight where there is an axis and vacuous where there is not. The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — the bound every account of the subject quotes. What occurs is one over the exponent of the cohomology, which divides the bound. For a rotation with a direction it fixes the two agree exactly: a four-fold screw does need quarters and a six-fold sixths. For a rotation acting with no fixed direction the exponent is one — the cohomology is trivial and no fraction occurs at all — so the bound is slack by the whole order. The same bound is sharp and useless in the same table.

The denominator a group actually needs

The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — a bound every account of the subject quotes. What occurs is one over the exponent, which divides it. The same bound turns out to be attained exactly and to be slack by its whole size, in two rows of one table, and what decides which is whether the rotation fixes a direction.

classification · Cohomology

Named alongside it

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

CocycleGroup extensionArithmetic crystal classClassificationIntrinsic translationOrigin shiftScrew axisConjugationCyclotomic polynomialsGlide planeHigher-dimensional latticeIndex

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