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tetraoctagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U84_1}}

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

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

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|+ Four uniform constructions of 4.8.4.8

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!Name

|Tetra-octagonal tiling

|Rhombi-octaoctagonal tiling

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!Image

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!Symmetry

|[8,4]
(*842)
{{CDD|node_c1|8|node_c2|4|node_c3}}

|[8,8] = [8,4,1+]
(*882)
{{CDD|node_c1|8|node_c2|4|node_h0}} = {{CDD|node_c1|split1-88|nodeab_c2}}

|[(4,4,4)] = [1+,8,4]
(*444)
{{CDD|node_h0|8|node_c2|4|node_c3}} = {{CDD|label4|branch_c2|split2-44|node_c3}}

|[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
{{CDD|node_h0|8|node_c2|4|node_h0}} = {{CDD|labelinfin|branch_c2|4a4b|branch_c2|labelinfin}} or {{CDD|nodeab_c2|4a4b-cross|nodeab_c2}}

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!Schläfli

|r{8,4}

|rr{8,8}
=r{8,4}1/2

|r(4,4,4)
=r{4,8}1/2

|t0,1,2,3(∞,4,∞,4)
=r{8,4}1/4

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!Coxeter

|{{CDD|node|8|node_1|4|node}}

|{{CDD|node|8|node_1|4|node_h0}} = {{CDD|node|split1-88|nodes_11}}

|{{CDD|node_h0|8|node_1|4|node}} = {{CDD|label4|branch_11|split2-44|node}}

|{{CDD|node_h0|8|node_1|4|node_h0}} = {{CDD|labelinfin|branch_11|4a4b|branch_11|labelinfin}} or {{CDD|nodes_11|4a4b-cross|nodes_11}}

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

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Related polyhedra and tiling

{{Quasiregular4 table}}

{{Quasiregular8 table}}

{{Order 8-4 tiling table}}

{{Order 8-8 tiling table}}

{{Order 4-4-4 tiling table}}

See also

{{Commons category|Uniform tiling 4-8-4-8}}

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 = 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]

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isotoxal tilings

Category:Uniform tilings