symmetry notation

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Patterns & Symmetry

Reading the symmetry notation

Every one of the 17 wallpaper groups carries a short code - p1, cmm, p4m, p6m - and those codes aren't arbitrary. They're international (Hermann-Mauguin) notation, the standard shorthand that crystallographers use for the two-dimensional symmetry groups, and they're the same labels Locus puts on its cluster presets and on the notation half of each group's diagram. Once you can read the code, the name alone tells you what mirrors, glides, and rotations a pattern has.

What the symbol encodes

A wallpaper-group symbol reads left to right as a short recipe:

1. Lattice letter - p or c. p is a primitive cell: the repeating parallelogram carries lattice points only at its corners. c is a centered cell: an extra lattice point sits at the middle, used when a rhombic pattern is cleaner to describe on a rectangle. Only two of the seventeen are centered - cm and cmm.
2. Rotation order - 1, 2, 3, 4, or 6. The highest order of rotational symmetry in the pattern. 1 means none (translations only); 2 is a half-turn, 4 a quarter-turn, and so on. The crystallographic restriction is why 5-fold and 7-fold never appear - no such rotation can tile the plane periodically.
3. Direction symbols - m, g, or 1. Each remaining position stands for a lattice direction, and its symbol says what reflection sits perpendicular to that direction: m a mirror (reflection) line, g a glide (reflect-and-slide) line, 1 nothing at all.

So the symbol names, in order, the cell type, the strongest rotation, and then the reflections and glides direction by direction.

Reading a symbol, position by position

• p4m: primitive cell, 4-fold rotation, mirror lines. The full form p4mm spells out that the mirrors run in two distinct directions: along the cell edges and along the diagonals.
• pmg: primitive, 2-fold, with a mirror in one direction and a glide in the perpendicular one. Full form p2mg.
• pgg: primitive, 2-fold, glides both ways and no mirrors at all. Full form p2gg.
• p3m1 vs p31m: both are primitive with 3-fold rotation and mirrors, but the position of the m and the 1 records where those mirrors sit relative to the lattice. In p3m1 every 3-fold center lies on a mirror line; in p31m the mirrors run along the cell edges and some 3-fold centers sit off them. Same ingredients, different arrangement - and the notation is the only part of the name that tells them apart.

That last pair is exactly why the direction positions matter: m1 and 1m are not the same group.

Short form vs full form

The codes on the wallpaper-groups page are the short form - the everyday names. Each also has a full form that writes out every position explicitly:

Short	Full	Short	Full
p1	p1	cmm	c2mm
p2	p211	p4	p4
pm	p1m1	p4m	p4mm
pg	p1g1	p4g	p4gm
cm	c1m1	p3	p3
pmm	p2mm	p3m1	p3m1
pmg	p2mg	p31m	p31m
pgg	p2gg	p6	p6
		p6m	p6mm

The short form simply drops 1s and ms that can be deduced, as long as no ambiguity remains - p1m1 collapses to pm, p2mm to pmm. p3m1 and p31m keep their 1s precisely because that is what distinguishes them.

The master chart

The definitive one-page reference is Chart 2 from Doris Schattschneider's 1978 paper, The Plane Symmetry Groups: Their Recognition and Notation. It draws all 17 lattice cells with their symmetry elements and gives both the short and full international notation for each - the source the Locus notation diagrams follow.

Chart 2 from Schattschneider (1978), p. 442. Click the chart to download the full paper (PDF). Reproduced for reference.

Reading the chart: double lines are mirror (reflection) axes, dashed lines are glide axes, and the small open shapes mark rotation centers - diamond = 2-fold, triangle = 3-fold, square = 4-fold, hexagon = 6-fold. The dotted outline on cm and cmm marks the "centered cell." These are the same conventions used in the symmetry-elements figure on each lattice page.

Why this paper

Before Schattschneider, the international (Hermann-Mauguin) notation lived mostly in the crystallography literature, where it had grown up to describe three-dimensional crystals and was awkward to apply to flat, two-dimensional patterns. Her paper did two things that made it a standard reference - it still turns up in the bibliographies of the field, from The American Mathematical Monthly to MathWorld:

• It laid the full and short international symbols side by side for all 17 groups in one clear chart, reconciling the competing notations then in use and settling on the names that mathematicians, tiling theorists, and designers have used ever since: the very codes on the wallpaper-groups page.
• It gave a practical recognition procedure: a step-by-step way to identify which of the 17 groups a given periodic pattern belongs to, by asking what its highest rotation is, whether it has reflections, and whether it has glides. The decision-table approach reproduced in countless later texts (and on Wikipedia today) descends from this.

That combination - a notation you can read and a method you can follow - is why the paper matters when you analyze wallpaper symmetry: it turns "which of the seventeen is this?" from an expert's judgment call into something anyone can work out and name unambiguously.

Citation. Doris Schattschneider, "The Plane Symmetry Groups: Their Recognition and Notation," The American Mathematical Monthly 85, no. 6 (June-July 1978): 439-450.

How it maps to what you see in Locus

Every group's symmetry-elements figure - the notation view on its lattice page - draws exactly these elements: mirror lines as double lines, glide lines dashed, and rotation centers as the hexagon / square / triangle / diamond glyphs. The code, the chart, and the figure all describe the same symmetry three ways. The cluster itself hands you the tiling and its fundamental domain; the domain's corners are the cell vectors, so the symmetry sits exactly where the figure says it does.

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Related

• The 17 wallpaper groups: every group as a cluster preset, with its tiling, cluster node, and notation diagram.