Tetrahexagonal tiling

Tetrahexagonal tiling
Tetrahexagonal tiling Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.6)2
Schläfli symbol	r{6,4} or $\left\{\begin{array}{c}6\\ 4\end{array}\right\}$ rr{6,6} r(4,4,3) t0,1,2,3(∞,3,∞,3)
Wythoff symbol	2 | 6 4
Coxeter diagram	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1-64.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-66.pngFile:CDel nodes 11.png File:CDel branch 11.pngFile:CDel split2-44.pngFile:CDel node.png File:CDel nodes 11.pngFile:CDel 3a3b-cross.pngFile:CDel nodes 11.png
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).

Four uniform constructions of 4.6.4.6

Uniform Coloring	File:H2 tiling 246-2.png	File:H2 tiling 266-5.png	File:H2 tiling 344-5.png	File:3222-uniform tiling-verf4646.png
Fundamental Domains	File:642 symmetry 000.png	File:642 symmetry 00a.png	File:642 symmetry a00.png	File:642 symmetry a0b.png
Schläfli	r{6,4}	r{4,6}1⁄2	r{6,4}1⁄2	r{6,4}1⁄4
Symmetry	[6,4] (*642) File:CDel node c3.pngFile:CDel 6.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.png	[6,6] = [6,4,1+] (*662) File:CDel node c3.pngFile:CDel split1-66.pngFile:CDel nodeab c1.png	[(4,4,3)] = [1+,6,4] (*443) File:CDel branch c1.pngFile:CDel split2-44.pngFile:CDel node c2.png	[(∞,3,∞,3)] = [1+,6,4,1+] (*3232) File:CDel labelinfin.pngFile:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c1.pngFile:CDel labelinfin.png or File:CDel nodeab c1.pngFile:CDel 3a3b-cross.pngFile:CDel nodeab c1.png
Symbol	r{6,4}	rr{6,6}	r(4,3,4)	t0,1,2,3(∞,3,∞,3)
Coxeter diagram	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel node.pngFile:CDel split1-66.pngFile:CDel nodes 11.png	File:CDel node h0.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png = File:CDel branch 11.pngFile:CDel split2-44.pngFile:CDel node.png	File:CDel node h0.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.png = File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.pngFile:CDel labelinfin.png or File:CDel nodes 11.pngFile:CDel 3a3b-cross.pngFile:CDel nodes 11.png

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.

File:Hyperbolic domains 3232.pngFile:Ord64 qreg rhombic til.pngFile:H2chess 246a.pngFile:Order-6 hexagonal tiling and dual.png

Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)2
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]
Figures	File:Uniform tiling 432-t1.png	File:Uniform tiling 44-t1.png	File:H2-5-4-rectified.svg	File:H2 tiling 246-2.png	File:H2 tiling 247-2.png	File:H2 tiling 248-2.png	File:H2 tiling 24i-2.png
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)2
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	File:Uniform tiling 63-t1.svg 6.3.6.3	File:H2 tiling 246-2.png 6.4.6.4	File:H2 tiling 256-2.png 6.5.6.5	File:H2 tiling 266-2.png 6.6.6.6	File:H2 tiling 267-2.png 6.7.6.7	File:H2 tiling 268-2.png 6.8.6.8	File:H2 tiling 26i-2.png 6.∞.6.∞	6.∞.6.∞
Dual figures
Rhombic figures configuration	File:Rhombic star tiling.svg V6.3.6.3	File:H2chess 246a.png V6.4.6.4	File:Order-6-5 quasiregular rhombic tiling.png V6.5.6.5	File:H2 tiling 246-4.png V6.6.6.6	V6.7.6.7	File:H2chess 268a.png V6.8.6.8	File:H2chess 26ia.png V6.∞.6.∞

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

References

• 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.

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