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Tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.6)
Schläfli symbol	r{6,4} or ${\displaystyle {\begin{Bmatrix}6\\4\end{Bmatrix}}}$ rr{6,6} r(4,4,3) t0,1,2,3(∞,3,∞,3)
Wythoff symbol	2 | 6 4
Coxeter diagram	or or
Symmetry group	, (*642) , (*662) , (*443) , (*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 kaleidoscope. Removing the last mirror, , gives , (*662). Removing the first mirror , gives , (*443). Removing both mirror as , leaving (*3232).

Four uniform constructions of 4.6.4.6

Uniform Coloring
Fundamental Domains
Schläfli	r{6,4}	r{4,6}1⁄2	r{6,4}1⁄2	r{6,4}1⁄4
Symmetry	(*642)	= (*662)	= (*443)	= (*3232) or
Symbol	r{6,4}	rr{6,6}	r(4,3,4)	t0,1,2,3(∞,3,∞,3)
Coxeter diagram		=	=	= or

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

*n42 symmetry mutations of quasiregular tilings: (4.n) v t e
Symmetry *4n2	Spherical	Euclidean	Compact hyperbolic				Paracompact	Noncompact
	*342	*442	*542	*642	*742	*842 ...	*∞42
Figures
Config.	(4.3)	(4.4)	(4.5)	(4.6)	(4.7)	(4.8)	(4.∞)	(4.ni)

Symmetry mutation of quasiregular tilings: (6.n) v t e
Symmetry *6n2	Euclidean	Compact hyperbolic					Paracompact	Noncompact
	*632	*642	*652	*662	*762	*862 ...	*∞62
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 v t e
Symmetry: , (*642) (with (*662), (*443) , (*3222) index 2 subsymmetries) (And (*3232) index 4 subsymmetry)
= = =	=	= = =	=	= = =	=
{6,4}	t{6,4}	r{6,4}	t{4,6}	{4,6}	rr{6,4}	tr{6,4}
Uniform duals
V6	V4.12.12	V(4.6)	V6.8.8	V4	V4.4.4.6	V4.8.12
Alternations
(*443)	(6*2)	(*3222)	(4*3)	(*662)	(2*32)	(642)
=	=	=	=	=	=
h{6,4}	s{6,4}	hr{6,4}	s{4,6}	h{4,6}	hrr{6,4}	sr{6,4}

Uniform hexahexagonal tilings v t e
Symmetry: , (*662)
= =	= =	= =	= =	= =	= =	= =
{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
V6	V6.12.12	V6.6.6.6	V6.12.12	V6	V4.6.4.6	V4.12.12
Alternations
(*663)	(6*3)	(*3232)	(6*3)	(*663)	(2*33)	(662)
=		=		=
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 v t e
Symmetry: (*443)							(443)	(3*22)	(*3232)
h{6,4} t0(4,4,3)	h2{6,4} t0,1(4,4,3)	{4,6}/2 t1(4,4,3)	h2{6,4} t1,2(4,4,3)	h{6,4} t2(4,4,3)	r{6,4}/2 t0,2(4,4,3)	t{4,6}/2 t0,1,2(4,4,3)	s{4,6}/2 s(4,4,3)	hr{4,6}/2 hr(4,3,4)	h{4,6}/2 h(4,3,4)	q{4,6} h1(4,3,4)
Uniform duals
V(3.4)	V3.8.4.8	V(4.4)	V3.8.4.8	V(3.4)	V4.6.4.6	V6.8.8	V3.3.3.4.3.4	V(4.4.3)	V6	V4.3.4.6.6

Similar H2 tilings in *3232 symmetry v t e
Coxeter diagrams
Vertex figure	6		(3.4.3.4)		3.4.6.6.4		6.4.6.4
Image
Dual

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

v t e Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff
Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet
Other Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg
By vertex type Spherical 2 3.n V3.n 4.n V4.n Regular 2 3 4 6 Semi- regular 3.4.3.4 V3.4.3.4 3.4 3.∞ 3.6 V3.6 3.4.6.4 (3.6) 3.12 4.∞ 4.6.12 4.8 Hyper- bolic 3.4.3.5 3.4.3.6 3.4.3.7 3.4.3.8 3.4.3.∞ 3.5.3.5 3.5.3.6 3.6.3.6 3.6.3.8 3.7.3.7 3.8.3.8 3.4.3.4 3.∞.3.∞ 3.7 3.8 3.∞ 3.4 3 3 3 (3.4) (3.4) 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7) (3.8) 3.14 3.16 3.∞ 4.5.4 4.6.4 4.7.4 4.8.4 4.∞.4 4 4 4 4 4 (4.5) (4.6) 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7) (4.8) 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.10 4.10.12 4.12 4.12.16 4.14 4.16 4.∞ 5 5 5 5 5.4.6.4 (5.6) 5.8 5.10 5.12 6 6 6 6 6.4.8.4 (6.8) 6.8 6.10 6.12 6.16 7 7 7 7.6 7.8 7.14 8 8 8 8 8.6 8.12 8.16 ∞ ∞ ∞ ∞ ∞.6 ∞.8

• Hyperbolic tilings
• Isogonal tilings
• Isotoxal tilings
• Uniform tilings