Tetraoctagonal tiling – Wikipedia
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Tetraoctagonal tiling | |
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Poincaré disk model of the hyperbolic plane |
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Type | Hyperbolic uniform tiling |
Vertex configuration | (4.8)2 |
Schläfli symbol | r{8,4} or rr{8,8} rr(4,4,4) t0,1,2,3(∞,4,∞,4) |
Wythoff symbol | 2 | 8 4 |
Coxeter diagram | or or |
Symmetry group | [8,4], (*842) [8,8], (*882) [(4,4,4)], (*444) [(∞,4,∞,4)], (*4242) |
Dual | Order-8-4 quasiregular rhombic tiling |
Properties | Vertex-transitive edge-transitive |
In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions[edit]
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Name | Tetra-octagonal tiling | Rhombi-octaoctagonal tiling | ||
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Image | ||||
Symmetry | [8,4] (*842) |
[8,8] = [8,4,1+] (*882) = |
[(4,4,4)] = [1+,8,4] (*444) = |
[(∞,4,∞,4)] = [1+,8,4,1+] (*4242) = or |
Schläfli | r{8,4} | rr{8,8} =r{8,4}1/2 |
r(4,4,4) =r{4,8}1/2 |
t0,1,2,3(∞,4,∞,4) =r{8,4}1/4 |
Coxeter | = | = | = or |
Symmetry[edit]
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
Related polyhedra and tiling[edit]
Uniform octagonal/square tilings | |||||||||||
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[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) |
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{8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} | |||||
Uniform duals | |||||||||||
V84 | V4.16.16 | V(4.8)2 | V8.8.8 | V48 | V4.4.4.8 | V4.8.16 | |||||
Alternations | |||||||||||
[1+,8,4] (*444) |
[8+,4] (8*2) |
[8,1+,4] (*4222) |
[8,4+] (4*4) |
[8,4,1+] (*882) |
[(8,4,2+)] (2*42) |
[8,4]+ (842) |
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h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
Alternation duals | |||||||||||
V(4.4)4 | V3.(3.8)2 | V(4.4.4)2 | V(3.4)3 | V88 | V4.44 | V3.3.4.3.8 |
Uniform octaoctagonal tilings | |||||||||||
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Symmetry: [8,8], (*882) | |||||||||||
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{8,8} | t{8,8} | r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
Uniform duals | |||||||||||
V88 | V8.16.16 | V8.8.8.8 | V8.16.16 | V88 | V4.8.4.8 | V4.16.16 | |||||
Alternations | |||||||||||
[1+,8,8] (*884) |
[8+,8] (8*4) |
[8,1+,8] (*4242) |
[8,8+] (8*4) |
[8,8,1+] (*884) |
[(8,8,2+)] (2*44) |
[8,8]+ (882) |
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= | = | = | = = |
= = |
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h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
Alternation duals | |||||||||||
V(4.8)8 | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)8 | V46 | V3.3.8.3.8 |
Uniform (4,4,4) tilings | |||||||||||
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Symmetry: [(4,4,4)], (*444) | [(4,4,4)]+ (444) |
[(1+,4,4,4)] (*4242) |
[(4+,4,4)] (4*22) |
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t0(4,4,4) h{8,4} |
t0,1(4,4,4) h2{8,4} |
t1(4,4,4) {4,8}1/2 |
t1,2(4,4,4) h2{8,4} |
t2(4,4,4) h{8,4} |
t0,2(4,4,4) r{4,8}1/2 |
t0,1,2(4,4,4) t{4,8}1/2 |
s(4,4,4) s{4,8}1/2 |
h(4,4,4) h{4,8}1/2 |
hr(4,4,4) hr{4,8}1/2 |
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Uniform duals | |||||||||||
V(4.4)4 | V4.8.4.8 | V(4.4)4 | V4.8.4.8 | V(4.4)4 | V4.8.4.8 | V8.8.8 | V3.4.3.4.3.4 | V88 | V(4,4)3 |
See also[edit]
References[edit]
External links[edit]
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