Crystallography & Symmetry
patterns derived from their groups
Generator

p4: 4 cosets, counted without a lattice

p4: 4 cosets, counted without a lattice
p4: 4 cosets, counted without a lattice. The finished Todd–Coxeter table for p4 with the subgroup generated by the two translations. Rows are cosets, columns are the generators and their inverses, and an entry says which coset a generator carries a coset to. The translations fix every row, which is what it means for them to be inside the subgroup; the other letters permute the rows, and the number of rows is 4 — the order of the point group of p4, arrived at from 4 words in 3 letters with no matrix, no lattice and no plane anywhere in the computation.

The finished Todd–Coxeter table for p4 with the subgroup generated by the two translations. Rows are cosets, columns are the generators and their inverses, and an entry says which coset a generator carries a coset to. The translations fix every row, which is what it means for them to be inside the subgroup; the other letters permute the rows, and the number of rows is 4 — the order of the point group of p4, arrived at from 4 words in 3 letters with no matrix, no lattice and no plane anywhere in the computation.

2 essays call coset-table. 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.

p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group. Operations

A group in four letters

Every other essay here describes a symmetry group by what it does to the plane. There is a second description — a handful of letters and the words in them that are required to equal nothing — and it can be counted with no plane anywhere in the computation.

 The ball of radius 5 in p6. Every element of p6 reachable in at most 5 multiplications by a generator or its inverse, plotted at its translation part — so each dot is a lattice position and its size says how few steps reach it. The picture is the word metric's unit ball scaled up, and its shape is what fixes the growth: a diamond where the group supplies two short translations, and a hexagon where it supplies three. Every dot here required the word problem to be solved, because the search has to know when two products are the same element. Operations

Telling two words apart

There are finitely presented groups in which no algorithm can decide whether two products of the generators are the same element. The seventeen are not among them, and the procedure that settles it is short enough to state in a sentence — which then makes it possible to measure how fast each group grows.

The whole library · All essays