Crystallography & Symmetry
patterns derived from their groups
Generator

pmm as a layer: two-sided

pmm as a layer: two-sided
pmm as a layer: two-sided. The plane group pmm, drawn as a layer in space. Every operation either leaves the two faces of the layer where they are or exchanges them, and the copies drawn in the second colour are the ones that turn it over — 4 of the 8 operations here. Looked at from above, all three of pmm's possible layers give the same picture of dots; what differs is which way up each copy of the motif is, and that is a fact about a solid rather than about a pattern. There are 5 such arrangements over pmm, and the choice between them is a homomorphism onto {±1} — the same arithmetic that colours a pattern in two colours, with 'the other side of the sheet' in place of 'the other colour'.

The plane group pmm, drawn as a layer in space. Every operation either leaves the two faces of the layer where they are or exchanges them, and the copies drawn in the second colour are the ones that turn it over — 4 of the 8 operations here. Looked at from above, all three of pmm's possible layers give the same picture of dots; what differs is which way up each copy of the motif is, and that is a fact about a solid rather than about a pattern. There are 5 such arrangements over pmm, and the choice between them is a homomorphism onto {±1} — the same arithmetic that colours a pattern in two colours, with 'the other side of the sheet' in place of 'the other colour'.

2 essays call layer-groups. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. The classification

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.

 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. The classification

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.

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