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order-4 pentagonal tiling

{{Short description|Regular tiling of the hyperbolic plane}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U54_0}}

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.

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Related polyhedra and tiling

{{Order 5-4 tiling table}}

{{Order_5-5_tiling_table}}

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 {{CDD|node_1|5|node|n|node}}, progressing to infinity.

class="wikitable collapsible collapsed"

!colspan=5| {5,n} tilings

align=center

|100px
{5,3}
{{CDD|node_1|5|node|3|node}}

|100px
{5,4}
{{CDD|node_1|5|node|4|node}}

|100px
{5,5}
{{CDD|node_1|5|node|5|node}}

|100px
{5,6}
{{CDD|node_1|5|node|6|node}}

|100px
{5,7}
{{CDD|node_1|5|node|7|node}}

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 {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.

{{Order-4_regular_tilings}}

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

{{Regular square tiling table}}

{{Quasiregular5 table}}

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • {{citation|first=H. S. M.|last=Coxeter|author-link= H. S. M. Coxeter|series=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|title= Chapter 10: Regular honeycombs in hyperbolic space|

url=http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf}}, invited lecture, ICM, Amsterdam, 1954.

See also

{{Commonscat|Order-4 pentagonal tiling}}

External links

  • {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
  • {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
  • [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
  • [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
  • [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isohedral tilings

Category:Order-4 tilings

Category:Pentagonal tilings

Category:Regular tilings

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