Order-4 pentagonal tiling

Template:Short description Template:Reg hyperbolic tiling stat table 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.

File:Uniform tiling 552-t1.png

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 Template:CDD, 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 Template:CDD, with n progressing to infinity. Template:Order-4 regular tilings

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

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn (Chapter 19, The Hyperbolic Archimedean Tessellations)
• , invited lecture, ICM, Amsterdam, 1954.

See also

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

External links

• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

Template:Hyperbolic-geometry-stub