Tetrapentagonal tiling

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t₁{4,5} or r{4,5}.

Tetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.5)2
Schläfli symbol	r{5,4} or ${\displaystyle {\begin{Bmatrix}5\\4\end{Bmatrix}}}$ rr{5,5} or ${\displaystyle r{\begin{Bmatrix}5\\5\end{Bmatrix}}}$
Wythoff symbol	2 | 5 4 5 5 | 2
Coxeter diagram	or or
Symmetry group	[5,4], (*542) [5,5], (*552)
Dual	Order-5-4 rhombille tiling
Properties	Vertex-transitive edge-transitive

Symmetry

A half symmetry [1⁺,4,5] = [5,5] construction exists, which can be seen as two colors of pentagons. This coloring can be called a rhombipentapentagonal tiling.

Dual tiling

The dual tiling is made of rhombic faces and has a face configuration V4.5.4.5:

Related polyhedra and tiling

Uniform pentagonal/square tilings v t e
Symmetry: [5,4], (*542)							[5,4]+, (542)	[5+,4], (5*2)	[5,4,1+], (*552)
{5,4}	t{5,4}	r{5,4}	2t{5,4}=t{4,5}	2r{5,4}={4,5}	rr{5,4}	tr{5,4}	sr{5,4}	s{5,4}	h{4,5}
Uniform duals
V54	V4.10.10	V4.5.4.5	V5.8.8	V45	V4.4.5.4	V4.8.10	V3.3.4.3.5	V3.3.5.3.5	V55

Uniform pentapentagonal tilings v t e
Symmetry: [5,5], (*552)							[5,5]+, (552)
=	=	=	=	=	=	=	=
Order-5 pentagonal tiling {5,5}	Truncated order-5 pentagonal tiling t{5,5}	Order-4 pentagonal tiling r{5,5}	Truncated order-5 pentagonal tiling 2t{5,5} = t{5,5}	Order-5 pentagonal tiling 2r{5,5} = {5,5}	Tetrapentagonal tiling rr{5,5}	Truncated order-4 pentagonal tiling tr{5,5}	Snub pentapentagonal tiling sr{5,5}
Uniform duals
Order-5 pentagonal tiling V5.5.5.5.5	V5.10.10	Order-5 square tiling V5.5.5.5	V5.10.10	Order-5 pentagonal tiling V5.5.5.5.5	V4.5.4.5	V4.10.10	V3.3.5.3.5

*n42 symmetry mutations of quasiregular tilings: (4.n)2 v t e
Symmetry *4n2 [n,4]	Spherical	Euclidean	Compact hyperbolic				Paracompact	Noncompact
	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]	[ni,4]
Figures
Config.	(4.3)2	(4.4)2	(4.5)2	(4.6)2	(4.7)2	(4.8)2	(4.∞)2	(4.ni)2

*5n2 symmetry mutations of quasiregular tilings: (5.n)2 v t e
Symmetry *5n2 [n,5]	Spherical	Hyperbolic					Paracompact	Noncompact
	*352 [3,5]	*452 [4,5]	*552 [5,5]	*652 [6,5]	*752 [7,5]	*852 [8,5]...	*∞52 [∞,5]	[ni,5]
Figures
Config.	(5.3)2	(5.4)2	(5.5)2	(5.6)2	(5.7)2	(5.8)2	(5.∞)2	(5.ni)2
Rhombic figures
Config.	V(5.3)2	V(5.4)2	V(5.5)2	V(5.6)2	V(5.7)2	V(5.8)2	V(5.∞)2	V(5.∞)2

See also

• Binary tiling, an aperiodic tiling of the hyperbolic plane by pentagons
• Uniform tilings in hyperbolic plane
• List of regular polytopes

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

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch