Infinite-order pentagonal tiling

Infinite-order pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	5^∞
Schläfli symbol	{5,∞}
Wythoff symbol	∞ \| 5 2
Coxeter diagram
Symmetry group	[∞,5], (*∞52)
Dual	Order-5 apeirogonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

Symmetry

There is a half symmetry form, , seen with alternating colors:

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5ⁿ).

Finite	Compact hyperbolic					Paracompact
{5,3}	{5,4}	{5,5}	{5,6}	{5,7}	{5,8}...	{5,∞}

Paracompact uniform apeirogonal/pentagonal tilings
Symmetry: [∞,5], (*∞52)							[∞,5]⁺ (∞52)	[1⁺,∞,5] (*∞55)		[∞,5⁺] (5*∞)


{∞,5}	t{∞,5}	r{∞,5}	2t{∞,5}=t{5,∞}	2r{∞,5}={5,∞}	rr{∞,5}	tr{∞,5}	sr{∞,5}	h{∞,5}	h₂{∞,5}	s{5,∞}
Uniform duals


V∞⁵	V5.∞.∞	V5.∞.5.∞	V∞.10.10	V5^∞	V4.5.4.∞	V4.10.∞	V3.3.5.3.∞		V(∞.5)⁵	V3.5.3.5.3.∞

John H. Conway; Heidi Burgiel; Chaim Goodman-Strass (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.

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