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:

Related polyhedra and tiling[]

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

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

Paracompact uniform apeirogonal/pentagonal tilings v
Symmetry: [∞,5], (*∞52)							[∞,5]+ (∞52)	[1+,∞,5] (*∞55)		[∞,5+] (5*∞)
{∞,5}		r{∞,5}		2r{∞,5}={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.∞

See also[]

Wikimedia Commons has media related to Infinite-order pentagonal tiling.

• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes

References[]

• 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.

External links[]

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery