Order-4 pentagonal tiling

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Order-4 pentagonal tiling
Order-4 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 54
Schläfli symbol {5,4}
r{5,5} or
Wythoff symbol 4 | 5 2
2 | 5 5
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png or CDel node 1.pngCDel split1-55.pngCDel nodes.png
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[edit]

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.

Uniform tiling 552-t1.png

Related polyhedra and tiling[edit]

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 CDel node 1.pngCDel 5.pngCDel node.pngCDel n.pngCDel node.png, 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 CDel node 1.pngCDel n.pngCDel node.pngCDel 4.pngCDel node.png, 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[edit]

See also[edit]

External links[edit]