Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

Tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.6)2
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	[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

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
Fundamental Domains
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) 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)2 v t e
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
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 v t e
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 v t e
Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*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
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)
=	=	=	=	=	=
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: [6,6], (*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
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)
=		=		=
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: [(4,4,3)] (*443)							[(4,4,3)]+ (443)	[(4,4,3+)] (3*22)	[(4,1+,4,3)] (*3232)
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
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 v t e
Coxeter diagrams
Vertex figure	66		(3.4.3.4)2		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