Truncated infinite-order triangular tiling

Infinite-order truncated triangular tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	∞.6.6
Schläfli symbol	t{3,∞}
Wythoff symbol	2 ∞ | 3
Coxeter diagram
Symmetry group	[∞,3], (*∞32)
Dual	apeirokis apeirogonal tiling
Properties	Vertex-transitive

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.

Symmetry

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

Small index subgroups of [(∞,3,3)], (*∞33)

Type	Reflectional	Rotational
Index	1	2
Diagram
Coxeter (orbifold)	[(∞,3,3)] (*∞33)	[(∞,3,3)]+ (∞33)

Related polyhedra and tiling

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6 v t e
Sym. *n42 [n,3]	Spherical				Euclid.	Compact		Parac.	Noncompact hyperbolic
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Truncated figures
Config.	2.6.6	3.6.6	4.6.6	5.6.6	6.6.6	7.6.6	8.6.6	∞.6.6	12i.6.6	9i.6.6	6i.6.6
n-kis figures
Config.	V2.6.6	V3.6.6	V4.6.6	V5.6.6	V6.6.6	V7.6.6	V8.6.6	V∞.6.6	V12i.6.6	V9i.6.6	V6i.6.6

Paracompact uniform tilings in [∞,3] family v t e
Symmetry: [∞,3], (*∞32)							[∞,3]+ (∞32)	[1+,∞,3] (*∞33)		[∞,3+] (3*∞)
		=	=	=				= or	= or	=
{∞,3}	t{∞,3}	r{∞,3}	t{3,∞}	{3,∞}	rr{∞,3}	tr{∞,3}	sr{∞,3}	h{∞,3}	h2{∞,3}	s{3,∞}
Uniform duals
V∞3	V3.∞.∞	V(3.∞)2	V6.6.∞	V3∞	V4.3.4.∞	V4.6.∞	V3.3.3.3.∞	V(3.∞)3		V3.3.3.3.3.∞

Paracompact hyperbolic uniform tilings in [(∞,3,3)] family v t e
Symmetry: [(∞,3,3)], (*∞33)							[(∞,3,3)]+, (∞33)
(∞,∞,3)	t0,1(∞,3,3)	t1(∞,3,3)	t1,2(∞,3,3)	t2(∞,3,3)	t0,2(∞,3,3)	t0,1,2(∞,3,3)	s(∞,3,3)
Dual tilings
V(3.∞)3	V3.∞.3.∞	V(3.∞)3	V3.6.∞.6	V(3.3)∞	V3.6.∞.6	V6.6.∞	V3.3.3.3.3.∞

See also

Wikimedia Commons has media related to Uniform tiling 6-6-i.

• 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.