Truncated tetraapeirogonal tiling

Truncated tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.8.∞
Schläfli symbol	tr{∞,4} or ${\displaystyle t{\begin{Bmatrix}\infty \\4\end{Bmatrix}}}$
Wythoff symbol	2 ∞ 4 |
Coxeter diagram	or
Symmetry group	[∞,4], (*∞42)
Dual	Order 4-infinite kisrhombille
Properties	Vertex-transitive

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

Paracompact uniform tilings in [∞,4] family v t e
{∞,4}	t{∞,4}	r{∞,4}	2t{∞,4}=t{4,∞}	2r{∞,4}={4,∞}	rr{∞,4}	tr{∞,4}
Dual figures
V∞4	V4.∞.∞	V(4.∞)2	V8.8.∞	V4∞	V43.∞	V4.8.∞
Alternations
[1+,∞,4] (*44∞)	[∞+,4] (∞*2)	[∞,1+,4] (*2∞2∞)	[∞,4+] (4*∞)	[∞,4,1+] (*∞∞2)	[(∞,4,2+)] (2*2∞)	[∞,4]+ (∞42)
=				=
h{∞,4}	s{∞,4}	hr{∞,4}	s{4,∞}	h{4,∞}	hrr{∞,4}	s{∞,4}
Alternation duals
V(∞.4)4	V3.(3.∞)2	V(4.∞.4)2	V3.∞.(3.4)2	V∞∞	V∞.44	V3.3.4.3.∞

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*242 [2,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Omnitruncated figure	4.8.4	4.8.6	4.8.8	4.8.10	4.8.12	4.8.14	4.8.16	4.8.∞
Omnitruncated duals	V4.8.4	V4.8.6	V4.8.8	V4.8.10	V4.8.12	V4.8.14	V4.8.16	V4.8.∞

*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n v t e
Symmetry *nn2 [n,n]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*222 [2,2]	*332 [3,3]	*442 [4,4]	*552 [5,5]	*662 [6,6]	*772 [7,7]	*882 [8,8]...	*∞∞2 [∞,∞]
Figure
Config.	4.4.4	4.6.6	4.8.8	4.10.10	4.12.12	4.14.14	4.16.16	4.∞.∞
Dual
Config.	V4.4.4	V4.6.6	V4.8.8	V4.10.10	V4.12.12	V4.14.14	V4.16.16	V4.∞.∞

The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1⁺,∞,1⁺,4,1⁺] (∞2∞2) is the commutator subgroup of [∞,4].

A larger subgroup is constructed as [∞,4*], index 8, as [∞,4⁺], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞⁴), and another [∞*,4], index ∞ as [∞⁺,4], (∞*2) with gyration points removed as (*2^∞). And their direct subgroups [∞,4*]⁺, [∞*,4]⁺, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2^∞).

Small index subgroups of [∞,4], (*∞42)
Index	1	2			4
Diagram
Coxeter	[∞,4]	[1+,∞,4] =	[∞,4,1+] =	[∞,1+,4] =	[1+,∞,4,1+] =	[∞+,4+]
Orbifold	*∞42	*∞44	*∞∞2	*∞222	*∞2∞2	∞2×
Semidirect subgroups
Diagram
Coxeter		[∞,4+]	[∞+,4]	[(∞,4,2+)]	[1+,∞,1+,4] = = = =	[∞,1+,4,1+] = = = =
Orbifold		4*∞	∞*2	2*∞2	∞*22	2*∞∞
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[∞,4]+ =	[∞,4+]+ =	[∞+,4]+ =	[∞,1+,4]+ =	[∞+,4+]+ = [1+,∞,1+,4,1+] = = =
Orbifold	∞42	∞44	∞∞2	∞222	∞2∞2
Radical subgroups
Index		8	∞	16	∞
Diagram
Coxeter		[∞,4*] =	[∞*,4]	[∞,4*]+ =	[∞*,4]+
Orbifold		*∞∞∞∞	*2∞	∞∞∞∞	2∞

Wikimedia Commons has media related to Uniform tiling 4-8-i.

• Tilings of regular polygons
• List of uniform planar tilings

• 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