Tetrahexagonal tiling
Appearance
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} or 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
[edit]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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Schläfli | r{6,4} | r{4,6}1⁄2 | r{6,4}1⁄2 | r{6,4}1⁄4 |
Symmetry | [6,4] (*642) ![]() ![]() ![]() ![]() ![]() |
[6,6] = [6,4,1+] (*662) ![]() ![]() ![]() |
[(4,4,3)] = [1+,6,4] (*443) ![]() ![]() ![]() |
[(∞,3,∞,3)] = [1+,6,4,1+] (*3232) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symbol | r{6,4} | rr{6,6} | r(4,3,4) | t0,1,2,3(∞,3,∞,3) |
Coxeter diagram |
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Symmetry
[edit]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
[edit]*n42 symmetry mutations of quasiregular tilings: (4.n)2 | ||||||||
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Symmetry *4n2 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||
*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 mutation of quasiregular tilings: 6.n.6.n | |||||||||||
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Symmetry *6n2 [n,6] |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||||||
*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.∞ |
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} |
Uniform hexahexagonal tilings | ||||||
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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} |
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 |
Similar H2 tilings in *3232 symmetry | ||||||||
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Coxeter diagrams |
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Vertex figure |
66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
Image | ![]() |
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Dual | ![]() |
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See also
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Uniform tiling 4-6-4-6.
References
[edit]- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.