Coxeter Groups
Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams:
For groups with whole number orders, including:
| Orbifold symmetry |
Coxeter group | Coxeter-Dynkin diagram |
notes | ||
|---|---|---|---|---|---|
| Compact | |||||
| *333 | (3 3 3) | ] | 3 reflective forms, 1 snub | ||
| *442 | (4 4 2) | 5 reflective forms, 1 snub | |||
| *632 | (6 3 2) | 7 reflective forms, 1 snub | |||
| *2222 | (∞ 2 ∞ 2) | × | 3 reflective forms, 1 snub | ||
| Noncompact | |||||
| *∞∞ | (∞) | ||||
| *22∞ | (2 2 ∞) | × | 2 reflective forms, 1 snub | ||
| Orbifold symmetry |
Coxeter group | Coxeter-Dynkin diagram |
notes | |
|---|---|---|---|---|
| Compact | ||||
| *pq2 | (p q 2) | 2(p+q) < pq | ||
| *pqr | (p q r) | pq+pr+qr < pqr | ||
| Noncompact | ||||
| *∞p2 | (p ∞ 2) | p>=3 | ||
| *∞pq | (p q ∞) | p,q>=3, p+q>6 | ||
| *∞∞p | (p ∞ ∞) | p>=3 | ||
| *∞∞∞ | (∞ ∞ ∞) | |||
Read more about this topic: Uniform Tiling
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