The affine Coxeter groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Coxeter group, and the Coxeter graph is obtained from the Coxeter graph of the Coxeter group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from An in this way, and the corresponding Coxeter group is the affine Weyl group of An. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
A list of the affine Coxeter groups follows:
| Group symbol |
Witt symbol |
Bracket notation | Related uniform tessellation(s) | Coxeter-Dynkin diagram |
|---|---|---|---|---|
| Pn+1 | ] | Simplectic honeycomb | ... or ... |
|
| Sn+1 | Demihypercubic honeycomb | ... | ||
| Rn+1 | Hypercubic honeycomb | ... | ||
| Qn+1 | Demihypercubic honeycomb | ... | ||
| T7 | 222 | |||
| T8 | 331, 133 | |||
| T9 | 521, 251, 152 | |||
| U5 | 16-cell honeycomb 24-cell honeycomb |
|||
| V3 | Hexagonal tiling and Triangular tiling |
|||
| W2 | apeirogon |
The subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.
Read more about this topic: Coxeter Group
Famous quotes containing the word groups:
“... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
—Mary Barnett Gilson (1877–?)