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hexaoctagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U86_1}}

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,6,1+], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1+,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1+,6,1+], leaves remaining mirrors (*4343).

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|+ Four uniform constructions of 6.8.6.8

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!Uniform
Coloring

|100px

|100px

|100px

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!Symmetry

|[8,6]
(*862)
{{CDD|node_c3|8|node_c1|6|node_c2}}

|[(8,3,8)] = [8,6,1+]
(*883)
{{CDD|node_c3|split1-88|branch_c1}}

|[(6,4,6)] = [1+,8,6]
(*664)
{{CDD|label4|branch_c1|split2-66|node_c2}}

|[1+,8,6,1+]
(*4343)
{{CDD|branch_c1|4a4b-cross|branch_c1}}

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!Symbol

|r{8,6}

|r{(8,3,8)}

|r{(6,4,6)}

|

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!Coxeter
diagram

|{{CDD|node|8|node_1|6|node}}

|{{CDD|node|8|node_1|6|node_h0}} = {{CDD|node|split1-88|branch_11}}

|{{CDD|node_h0|8|node_1|6|node}} = {{CDD|branch_11|split2-66|node}}

|{{CDD|node_h0|8|node_1|6|node_h0}} =
{{CDD|branch_11|4a4b-cross|branch_11}}

Symmetry

The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6].

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[1+,8,4,1+], (*4343)

!160px
[(8,4,2+)], (2*43)

Related polyhedra and tiling

{{Order 8-6 tiling table}}

{{Quasiregular6 table}}

{{Quasiregular8 table}}

See also

{{Commonscat|Uniform tiling 6-8-6-8}}

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)
  • {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

External links

  • {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
  • {{MathWorld | urlname=PoincareHyperbolicDisk | title = PoincarĂ© hyperbolic disk }}
  • [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery] {{Webarchive|url=https://web.archive.org/web/20130324095520/http://bork.hampshire.edu/~bernie/hyper/ |date=2013-03-24 }}
  • [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]

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Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isotoxal tilings

Category:Uniform tilings