Crystallography & Symmetry
patterns derived from their groups
Concept

Axial class — where it appears

A crystal class that leaves some direction invariant, which is every class but the five cubic ones. They are the classes a layer or a rod may have, since both objects single out a direction and a class with four three-fold axes cannot.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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. The seventy-five rod groups, the eighty layer groups and the two hundred and thirty space groups are numbers from the literature, marked as such wherever they appear: reaching them needs the translation extensions and their equivalences in full, which is the content of the classification rather than an application of it. The subperiodic cases sit exactly between the two halves, which is why they are so easy to assume are already known.

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.

classification · Subperiodic 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.

classification · Subperiodic The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups

Named alongside it

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

EnumerationFrieze groupHomomorphismLayer groupRod groupCountingCrystal classCrystallographic restrictionGlide reflectionHelixHermann–Mauguin notationIntrinsic translation

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