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

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as , removing two of three mirrors (passing through the pentagon center) in the symmetry.

The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t₁{5,5} and as a quasiregular tiling is called a pentapentagonal tiling.

Related polyhedra and tiling

Uniform pentagonal/square tilings v t e
Symmetry: , (*542)							, (542)	, (5*2)	, (*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
V5	V4.10.10	V4.5.4.5	V5.8.8	V4	V4.4.5.4	V4.8.10	V3.3.4.3.5	V3.3.5.3.5	V5

Uniform pentapentagonal tilings v t e
Symmetry: , (*552)							, (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

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol {5,n}, and Coxeter diagram , progressing to infinity.

{5,n} tilings
{5,3}	{5,4}	{5,5}	{5,6}	{5,7}

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

*n42 symmetry mutation of regular tilings: {n,4} v t e
Spherical		Euclidean	Hyperbolic tilings
2	3	4	5	6	7	8	...∞

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4).

*n42 symmetry mutation of regular tilings: {4,n} v t e
Spherical	Euclidean	Compact hyperbolic				Paracompact
{4,3}	{4,4}	{4,5}	{4,6}	{4,7}	{4,8}...	{4,∞}

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

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)
• Coxeter, H. S. M. (1999), Chapter 10: Regular honeycombs in hyperbolic space (PDF), The Beauty of Geometry: Twelve Essays, Dover Publications, ISBN 0-486-40919-8, LCCN 99035678, invited lecture, ICM, Amsterdam, 1954.

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

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

v t e Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff
Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet
Other Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg
By vertex type Spherical 2 3.n V3.n 4.n V4.n Regular 2 3 4 6 Semi- regular 3.4.3.4 V3.4.3.4 3.4 3.∞ 3.6 V3.6 3.4.6.4 (3.6) 3.12 4.∞ 4.6.12 4.8 Hyper- bolic 3.4.3.5 3.4.3.6 3.4.3.7 3.4.3.8 3.4.3.∞ 3.5.3.5 3.5.3.6 3.6.3.6 3.6.3.8 3.7.3.7 3.8.3.8 3.4.3.4 3.∞.3.∞ 3.7 3.8 3.∞ 3.4 3 3 3 (3.4) (3.4) 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7) (3.8) 3.14 3.16 3.∞ 4.5.4 4.6.4 4.7.4 4.8.4 4.∞.4 4 4 4 4 4 (4.5) (4.6) 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7) (4.8) 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.10 4.10.12 4.12 4.12.16 4.14 4.16 4.∞ 5 5 5 5 5.4.6.4 (5.6) 5.8 5.10 5.12 6 6 6 6 6.4.8.4 (6.8) 6.8 6.10 6.12 6.16 7 7 7 7.6 7.8 7.14 8 8 8 8 8.6 8.12 8.16 ∞ ∞ ∞ ∞ ∞.6 ∞.8

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