Knowledge (XXG)

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}.

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

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.

*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.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

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

• 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.

• 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

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