Order-4 pentagonal tiling

Order-4 pentagonal tiling Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 54
Schläfli symbol {5,4}
r{5,5} or ${\begin{Bmatrix}5\\5\end{Bmatrix}}$ Wythoff symbol 4 | 5 2
2 | 5 5
Coxeter diagram          or   Symmetry group [5,4], (*542)
[5,5], (*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 [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.

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.

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).