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).
class=wikitable width=520
|+ Four uniform constructions of 4.8.4.8 |
align=center
!Name |Tetra-octagonal tiling |Rhombi-octaoctagonal tiling | | |
align=center
!Image |80px |80px |80px |80px |
align=center
|[8,4] |[8,8] = [8,4,1+] |[(4,4,4)] = [1+,8,4] |[(∞,4,∞,4)] = [1+,8,4,1+] |
align=center
|r{8,4} |rr{8,8} |r(4,4,4) |t0,1,2,3(∞,4,∞,4) |
align=center
|{{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.
160px |
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]