Truncated order-6 square tiling
| Truncated order-6 square tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 8.8.6 |
| Schläfli symbol | t{4,6} |
| Wythoff symbol | 2 6 | 4 |
| Coxeter diagram |    
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| Symmetry group | [6,4], (*642) [(3,3,4)], (*334) |
| Dual | Order-4 hexakis hexagonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,6}.
Symmetry
The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons:
Dual tiling
The dual tiling represents the fundamental domains of the *443 orbifold, shown in two different centers here:
Related polyhedra and tiling
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
| Uniform tetrahexagonal tilings | |||||||||||
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| Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
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| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
| Uniform duals | |||||||||||
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| V64 | V4.12.12 | V(4.6)2 | V6.8.8 | V46 | V4.4.4.6 | V4.8.12 | |||||
| Alternations | |||||||||||
| [1+,6,4] (*443) |
[6+,4] (6*2) |
[6,1+,4] (*3222) |
[6,4+] (4*3) |
[6,4,1+] (*662) |
[(6,4,2+)] (2*32) |
[6,4]+ (642) | |||||
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| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
It can also be generated from the (4 4 3) hyperbolic tilings:
| Uniform (4,4,3) tilings | ||||||||||
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| Symmetry: [(4,4,3)] (*443) | [(4,4,3)]+ (443) |
[(4,4,3+)] (3*22) |
[(4,1+,4,3)] (*3232) | |||||||
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| h{6,4} t0(4,4,3) |
h2{6,4} t0,1(4,4,3) |
{4,6}1/2 t1(4,4,3) |
h2{6,4} t1,2(4,4,3) |
h{6,4} t2(4,4,3) |
r{6,4}1/2 t0,2(4,4,3) |
t{4,6}1/2 t0,1,2(4,4,3) |
s{4,6}1/2 s(4,4,3) |
hr{4,6}1/2 hr(4,3,4) |
h{4,6}1/2 h(4,3,4) |
q{4,6} h1(4,3,4) |
| Uniform duals | ||||||||||
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| V(3.4)4 | V3.8.4.8 | V(4.4)3 | V3.8.4.8 | V(3.4)4 | V4.6.4.6 | V6.8.8 | V3.3.3.4.3.4 | V(4.4.3)2 | V66 | V4.3.4.6.6 |
| *n42 symmetry mutation of truncated tilings: n.8.8 | |||||||||||
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| Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
| *242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | ||||
| Truncated figures |
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| Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 | |||
| n-kis figures |
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| Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 | |||
See also
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.