order-4 hexagonal tiling
{{short description|Regular tiling of the hyperbolic plane}}
{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U64_0}}
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6*,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.
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Uniform colorings
There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.
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|+ Uniform constructions of 6.6.6.6 |
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! !1 color !2 colors !colspan=2|3 and 2 colors !colspan=3|4, 3 and 2 colors |
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|[6,4] |[6,6] |colspan=2|[(6,6,3)] = [6,6,1+] |colspan=3|[1+,6,6,1+] |
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!Symbol |{6,4} |r{6,6} = {6,4}1/2 |colspan=2|r(6,3,6) = r{6,6}1/2 |colspan=3|r{6,6}1/4 |
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|{{CDD|node_1|6|node|4|node}} |{{CDD|node_1|split1-66|nodes}} = {{CDD|node_1|6|node|4|node_h0}} |colspan=2|{{CDD|node|split1-66|branch_11}} = {{CDD|node|6|node_1|6|node_h0}} |colspan=3|{{CDD|branch_11|3a3b-cross|branch_11}} = {{CDD|node_h0|6|node_1|6|node_h0}} = {{CDD|node_1|6|node_g|4sg|node_g}} |
Regular maps
The regular map {6,4}3 or {6,4}(4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron, {3,4}{{pi}}, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram {{CDD|node_1|6|node|n|node}}, progressing to infinity.
{{Hexagonal_regular_tilings}}
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}}
{{Quasiregular6 table}}
{{Order 6-4 tiling table}}
{{Order 6-6 tiling table}}
{{Order 3-2-3-2 tiling table}}
{{Order_3-2-2-2_tiling_table}}
See also
{{Commons category|Order-4 hexagonal tiling}}
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]
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