Tetraoctagonal tiling

Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.8)²
Schläfli symbol	r{8,4} or ${\begin{Bmatrix}8\\4\end{Bmatrix}}$ rr{8,8} rr(4,4,4) t_0,1,2,3(∞,4,∞,4)
Wythoff symbol	2 \| 8 4
Coxeter diagram	or or
Symmetry group	[8,4], (842) [8,8], (882) [(4,4,4)], (444) [(∞,4,∞,4)], (4242)
Dual	Order-8-4 quasiregular rhombic tiling
Properties	Vertex-transitive edge-transitive

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1⁺], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1⁺,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1⁺,8,4,1⁺], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Four uniform constructions of 4.8.4.8
Name	Tetra-octagonal tiling	Rhombi-octaoctagonal tiling
Image
Symmetry	[8,4] (*842)	[8,8] = [8,4,1⁺] (*882) =	[(4,4,4)] = [1⁺,8,4] (*444) =	[(∞,4,∞,4)] = [1⁺,8,4,1⁺] (*4242) = or
Schläfli	r{8,4}	rr{8,8} =r{8,4}¹/₂	r(4,4,4) =r{4,8}¹/₂	t_0,1,2,3(∞,4,∞,4) =r{8,4}¹/₄
Coxeter		=	=	= or

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) 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
Symmetry *4n2 [n,4]	*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

Dimensional family of quasiregular polyhedra and tilings: (8.n)² v t e
Symmetry *8n2 [n,8]	Hyperbolic...						Paracompact	Noncompact
Symmetry *8n2 [n,8]	*832 [3,8]	*842 [4,8]	*852 [5,8]	*862 [6,8]	*872 [7,8]	*882 [8,8]...	*∞82 [∞,8]	[iπ/λ,8]
Coxeter
Quasiregular figures configuration	3.8.3.8	4.8.4.8	8.5.8.5	8.6.8.6	8.7.8.7	8.8.8.8	8.∞.8.∞	8.∞.8.∞

Uniform octagonal/square tilings v t e
[8,4], (842) (with [8,8] (882), [(4,4,4)] (444) , [∞,4,∞] (4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
= = =	=	= = =	=	= =	=

{8,4}	t{8,4}	r{8,4}	2t{8,4}=t{4,8}	2r{8,4}={4,8}	rr{8,4}	tr{8,4}
Uniform duals


V8⁴	V4.16.16	V(4.8)²	V8.8.8	V4⁸	V4.4.4.8	V4.8.16
Alternations
[1⁺,8,4] (*444)	[8⁺,4] (8*2)	[8,1⁺,4] (*4222)	[8,4⁺] (4*4)	[8,4,1⁺] (*882)	[(8,4,2⁺)] (2*42)	[8,4]⁺ (842)
=	=	=	=	=	=

h{8,4}	s{8,4}	hr{8,4}	s{4,8}	h{4,8}	hrr{8,4}	sr{8,4}
Alternation duals


V(4.4)⁴	V3.(3.8)²	V(4.4.4)²	V(3.4)³	V8⁸	V4.4⁴	V3.3.4.3.8

Uniform octaoctagonal tilings v t e
Symmetry: [8,8], (*882)
= =	= =	= =	= =	= =	= =	= =

{8,8}	t{8,8}	r{8,8}	2t{8,8}=t{8,8}	2r{8,8}={8,8}	rr{8,8}	tr{8,8}
Uniform duals


V8⁸	V8.16.16	V8.8.8.8	V8.16.16	V8⁸	V4.8.4.8	V4.16.16
Alternations
[1⁺,8,8] (*884)	[8⁺,8] (8*4)	[8,1⁺,8] (*4242)	[8,8⁺] (8*4)	[8,8,1⁺] (*884)	[(8,8,2⁺)] (2*44)	[8,8]⁺ (882)
=		=		=	= =	= =

h{8,8}	s{8,8}	hr{8,8}	s{8,8}	h{8,8}	hrr{8,8}	sr{8,8}
Alternation duals


V(4.8)⁸	V3.4.3.8.3.8	V(4.4)⁴	V3.4.3.8.3.8	V(4.8)⁸	V4⁶	V3.3.8.3.8

Uniform (4,4,4) tilings v t e
Symmetry: [(4,4,4)], (*444)							[(4,4,4)]⁺ (444)	[(1⁺,4,4,4)] (*4242)	[(4⁺,4,4)] (4*22)


t₀(4,4,4) h{8,4}	t_0,1(4,4,4) h₂{8,4}	t₁(4,4,4) {4,8}¹/₂	t_1,2(4,4,4) h₂{8,4}	t₂(4,4,4) h{8,4}	t_0,2(4,4,4) r{4,8}¹/₂	t_0,1,2(4,4,4) t{4,8}¹/₂	s(4,4,4) s{4,8}¹/₂	h(4,4,4) h{4,8}¹/₂	hr(4,4,4) hr{4,8}¹/₂
Uniform duals

V(4.4)⁴	V4.8.4.8	V(4.4)⁴	V4.8.4.8	V(4.4)⁴	V4.8.4.8	V8.8.8	V3.4.3.4.3.4	V8⁸	V(4,4)³

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.