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 $t{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Wythoff symbol	2 ∞ 4 \|
Coxeter diagram	or
Symmetry group	, (*∞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}.

Related polyhedra and tilings

Paracompact uniform tilings in family v t e


{∞,4}	t{∞,4}	r{∞,4}	2t{∞,4}=t{4,∞}	2r{∞,4}={4,∞}	rr{∞,4}	tr{∞,4}
Dual figures


V∞	V4.∞.∞	V(4.∞)	V8.8.∞	V4	V4.∞	V4.8.∞
Alternations
(*44∞)	(∞*2)	(*2∞2∞)	(4*∞)	(*∞∞2)	(2*2∞)	(∞42)
=				=
h{∞,4}	s{∞,4}	hr{∞,4}	s{4,∞}	h{4,∞}	hrr{∞,4}	s{∞,4}

Alternation duals


V(∞.4)	V3.(3.∞)	V(4.∞.4)	V3.∞.(3.4)	V∞	V∞.4	V3.3.4.3.∞

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n v t e
Symmetry *n42	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry *n42	*242	*342	*442	*542	*642	*742	*842 ...	*∞42
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	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry *nn2	*222	*332	*442	*552	*662	*772	*882 ...	*∞∞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.∞.∞

Symmetry

The dual of this tiling represents the fundamental domains of , (*∞42) symmetry. There are 15 small index subgroups constructed from 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, (∞2∞2) is the commutator subgroup of .

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

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

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

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

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