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Octagonal tiling

{{other uses|truncated square tiling}}

{{Technical|date=May 2021}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U83_0}}

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

Uniform colorings

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

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!Regular

!colspan=2|Truncations

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|150px
{8,3}
{{CDD|node_1|8|node|3|node}}

|150px
t{4,8}
{{CDD|node_1|4|node_1|8|node}}

|150px
t{4[3]}
{{CDD|node_1|8|node_g|3sg|node_g}} = {{CDD|node_1|4|node_1|8|node_h0}} = {{CDD|node_1|split1-44|branch_11|label4}}

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!colspan=4| Dual tiling

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|150px
{3,8}
{{CDD|node_f1|8|node|3|node}} = {{CDD|node|8|node|3|node_1}}

|150px
{{CDD|node|8|node_f1|3|node_f1}} = {{CDD|node_1|split1|branch|label4}}

|150px
{{CDD|node_f1|8|node_g|3sg|node_g}} = {{CDD|node_f1|4|node_f1|8|node_h0}} = {{CDD|3|node_f1|4|node_f1|4|node_f1|4}}

Regular maps

The regular map {8,3}2,0 can be seen as a 6-coloring of the {8,3} hyperbolic tiling. Within the regular map, octagons of the same color are considered the same face shown in multiple locations. The 2,0 subscripts show the same color will repeat by moving 2 steps in a straight direction following opposite edges. This regular map also has a representation as a double covering of a cube, represented by Schläfli symbol {8/2,3}, with 6 octagonal faces, double wrapped {8/2}, with 24 edges, and 16 vertices. It was described by Branko Grünbaum in his 2003 paper Are Your Polyhedra the Same as My Polyhedra?{{cite journal |last1=Grünbaum |first1=Branko |title=Are Your Polyhedra the Same as My Polyhedra? |journal=Discrete and Computational Geometry |date=2003 |volume=25 |pages=461–488 |doi=10.1007/978-3-642-55566-4_21 |url=http://faculty.washington.edu/moishe/branko/Your%20polyhedra,%20my%20polyhedra/grun.pdf |access-date=27 April 2023}}

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Related polyhedra and tilings

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

{{Order-3 tiling table}}

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

{{Octagonal tilings}}

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

{{Order 8-3 tiling table}}

{{Order 8-4 tiling table}}

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

See also

{{Commonscat|Order-3 octagonal tiling}}

References

{{reflist}}

{{refbegin}}

  • 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}}

{{refend}}

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:Regular tilings