Order-4 pentagonal tiling

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.

Order-4 pentagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration54
Schläfli symbol{5,4}
r{5,5} or
Wythoff symbol5 2
2 | 5 5
Coxeter diagram
or
Symmetry group[5,4], (*542)
[5,5], (*552)
DualOrder-5 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

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 [5*,4], removing two of three mirrors (passing through the pentagon center) in the [5,4] 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.

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.

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.

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

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)
  • 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
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