Uniform Tiling - Coxeter Groups

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:

Euclidean plane
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
Hyperbolic plane
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
*∞∞∞ (∞ ∞ ∞)

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Famous quotes containing the word groups:

    “Only the groups which exclude us have magic.”
    —Mason Cooley (b. 1927)