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
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