Tetrapentagonal tiling

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

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}.

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.

File:Uniform tiling 552-t02.png

Dual tiling

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

File:H2-5-4-rhombic.svg

Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)2
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	File:Uniform tiling 432-t1.svg	File:Uniform tiling 44-t1-1.svg	File:H2-5-4-rectified.svg	File:H2 tiling 246-2.png	File:H2 tiling 247-2.png	File:H2 tiling 248-2.png	File:H2 tiling 24i-2.png
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
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	File:Uniform tiling 532-t1.png	File:H2-5-4-rectified.svg	File:H2 tiling 255-2.png	File:H2 tiling 256-2.png	File:H2 tiling 257-2.png	File:H2 tiling 258-2.png	File:H2 tiling 25i-2.png
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	File:Rhombictriacontahedron.svg	File:H2-5-4-rhombic.svg	File:H2-5-4-primal.svg	File:Order-6-5 quasiregular rhombic tiling.png
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, Template:Isbn (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

• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

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