Tetrahexagonal tiling
| Tetrahexagonal tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.6)2 |
| Schläfli symbol | r{6,4} rr{6,6} r(4,4,3) t0,1,2,3{(∞,3,∞,3)} |
| Wythoff symbol | 2 | 6 4 |
| Coxeter diagram |         |
| Symmetry group | [6,4], (*642) [6,6], (*662) [(4,4,3)], (*443) [(∞,3,∞,3)], (*3232) |
| Dual | Order-6-4 quasiregular rhombic tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).
| Uniform Coloring |
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| Fundamental Domains |
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| Symmetry | [6,4] (*642)    |
[6,6] = [6,4,1+] (*662)   |
[(4,4,3)] = [1+,6,4] (*443)  |
[(∞,3,∞,3)] = [1+,6,4,1+] (*3232)   or |
| Symbol | r{6,4} | rr{6,6} | r(4,3,4) | t0,1,2,3(∞,3,∞,3) |
| Coxeter diagram |
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Symmetry
The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.
Related polyhedra and tiling
| Symmetry *4n2 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||
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| *342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] |
[ni,4] | |
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| Config. | (4.3)2 | (4.4)2 | (4.5)2 | (4.6)2 | (4.7)2 | (4.8)2 | (4.∞)2 | (4.ni)2 |
| Symmetry *6n2 [n,6] |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | ||||
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| *632 [3,6] |
*642 [4,6] |
*652 [5,6] |
*662 [6,6] |
*762 [7,6] |
*862 [8,6]... |
*∞62 [∞,6] |
[iπ/λ,6] | |
| Quasiregular figures configuration |
 6.3.6.3 |
6.4.6.4 |
 6.5.6.5 |
 6.6.6.6 |
 6.7.6.7 |
 6.8.6.8 |
 6.∞.6.∞ |
6.∞.6.∞ |
| Dual figures | ||||||||
| Rhombic figures configuration |
 V6.3.6.3 |
 V6.4.6.4 |
 V6.5.6.5 |
 V6.6.6.6 |
V6.7.6.7 |
 V6.8.6.8 |
 V6.∞.6.∞ |
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| Symmetry: [6,6], (*662) | ||||||
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| {6,6} = h{4,6} |
t{6,6} = h2{4,6} |
r{6,6} {6,4} |
t{6,6} = h2{4,6} |
{6,6} = h{4,6} |
rr{6,6} r{6,4} |
tr{6,6} t{6,4} |
| Uniform duals | ||||||
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| V66 | V6.12.12 | V6.6.6.6 | V6.12.12 | V66 | V4.6.4.6 | V4.12.12 |
| Alternations | ||||||
| [1+,6,6] (*663) |
[6+,6] (6*3) |
[6,1+,6] (*3232) |
[6,6+] (6*3) |
[6,6,1+] (*663) |
[(6,6,2+)] (2*33) |
[6,6]+ (662) |
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| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
{Hidden end}}
| 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)} {(4,4,3)} |
h2{6,4} t0,1{(4,4,3)} r{(3,4,4)} |
{4,6} t1{(4,4,3)} {(4,3,4)} |
h2{6,4} t1,2{(4,4,3)} r{(4,4,3)} |
h{6,4} t2{(4,4,3)} {(3,4,4)} |
r{6,4} t0,2{(4,4,3)} r{(4,3,4)} |
t{4,6} t0,1,2{(4,4,3)} t{(4,3,4)} |
s{4,6} s{(4,4,3)} |
hr{6,4} hr{(4,3,4)} |
h{4,6} h{(4,3,4)} |
q{4,6} h2{(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 |
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| Vertex figure |
66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
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See also
 |
Wikimedia Commons has media related to Uniform tiling 4-6-4-6. |
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.
External links
- Weisstein, Eric W., "Hyperbolic tiling", MathWorld.
- Weisstein, Eric W., "Poincaré hyperbolic disk", MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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