Order-4 pentagonal tiling

Order-4 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Regular hyperbolic tiling
Vertex figure	5.5.5.5
Schläfli symbol(s)	{5,4}
Wythoff symbol(s)	4 | 5 2
Coxeter-Dynkin(s)
Coxeter group	[5,4]
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}.

Related polyhedra and tiling

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.

{5,3}

{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.

{3,4}

{4,4}

{5,4}

{6,4}

{7,4}

...

{∞,4}

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space)

See also

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

External links

• Weisstein, Eric W., "Hyperbolic tiling" from MathWorld.
• Weisstein, Eric W., "Poincaré hyperbolic disk" from MathWorld.
• 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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