Subperiodic — the series · Crystallography & Symmetry

Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. So are both subperiodic families: the seventy-five rod groups class by class, and the eighty layer groups as seventeen one-sided, seventeen mirrored and the forty-six two-colour classes of the plane group underneath. The two hundred and thirty space groups remain a number from the literature, marked as such wherever it appears: reaching it needs the translation extensions of seventy-three arithmetic classes in full.

A layer is not a wallpaper

A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.

part 1 · classification

The symmetry of a cut through Pnma. Every height in one cell, and the number of operations of Pnma that map the plane at that height to itself. The answer is 2 almost everywhere and rises to 4 at the special heights, where the plane group named above the spike is what a reader looking down at that surface would see. The rule is two conditions and no more: the operation must not tilt the plane, and the plane must come back to its own height — so a twofold axis lying in the plane survives at two heights per cell and a screw axis along the normal survives nowhere.

What a cleave leaves

A surface is a crystal that has been cut, and the symmetry it presents is what the space group leaves of itself on that plane. Two conditions decide it — the plane must not tilt, and it must come back to its own height — and the answer changes with where the cut was made.

part 2 · classification

75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it.

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

part 3 · classification

The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis.

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

part 4 · classification

Eighty layer groups, over the seventeen plane groups. Each of the seventeen plane groups with a block for every layer group that projects onto it. Every plane group carries one layer whose two faces differ and nothing turns it over, and one with a mirror in its own plane. The rest are turned over by some of their operations, and there is one for each class of two-colouring of the plane group: the pale-dark blocks turn the layer over only with a rotation, a mirror line or a glide line of the pattern, and the dark blocks turn it over on a translation — a glide plane lying in the layer. The totals are 17, 17, 26 and 20, which is eighty.

Eighty is seventeen, seventeen and forty-six

The eighty layer groups are usually quoted, and an earlier count here reached sixty-three and blamed the translations. The eighty follow in three lines from things already counted: a layer group is a plane group with a sign on each operation, and the signs are either all plus, all doubled by a mirror in the layer, or a two-colouring of the plane group — seventeen, seventeen and forty-six. The sixty-three had missed the twenty colourings that turn a layer over on a translation, which are glide planes lying in the layer itself.

part 5 · classification

The seventy-five rod groups by what fixes the axis and what moves along it. Every rod group sorted by its kernel, the plane point group of operations that leave every point of the axis in place, and by the group of the line it maps onto: translations only, or translations and reversals. Each chip counts the rods with that kernel, that line group and a given number of steps to the repeat. 29 never reverse the axis and 46 do, seventy-five in all, derived from the kernel and the rule that joins it to the line and agreeing cell by cell with the rods counted class by class.

Seventy-five is twenty-nine and forty-six

The eighty layer groups came out as a plane group with a sign on each operation. The seventy-five rod groups have an account of the same shape turned round: a finite group of the plane across the rod, which is what leaves the axis alone, extended by a group of the line, which is what moves along it. Twenty-nine rods never turn their axis over and forty-six do, and the count comes out cell by cell as the rods counted class by class. The screw of a rod is the turn its smallest step gives the plane, and the kernel decides its subscript.

part 6 · classification

The two forty-sixes over the ten plane point groups. The forty-six two-colour plane groups by the point group of the uncoloured group, and the forty-six rods that turn their axis over by the plane point group their operations show end on. The colourings pile onto the rectangular mirror groups — seventeen on 2mm, ten on m, none on 3 — and the rods onto the hexagonal and square groups, eleven on 6mm and nine on 4mm. Seventeen of the forty-six would have to change point group for the two lists to match.

Two forty-sixes that do not match

The layer groups contain forty-six two-coloured plane patterns, and the rod groups contain forty-six threads that turn their axis over. Both lists are lists of a group with a half of index two, and both halves have a point group in the plane, so they can be laid side by side. They do not line up on any invariant tried: the plane spends its forty-six on rectangular mirrors, which halve a lattice easily, and the rods spend theirs on high rotations, which carry the most kinds of screw. The equal totals are two sums that happen to meet.

part 7 · classification