Octagonal tiling

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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.

Regular Truncations
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t12.png
t{4,8}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
Uniform tiling 444-t012.png
t{4[3]}
CDel node 1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.png
Dual tiling
H2-8-3-primal.svg
{3,8}
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 433-t2.png
CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png = CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2checkers 444.png
CDel node f1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node h0.png = CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png

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?[1]

Double-cube-regular-map.png

Related polyhedra and tilings

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}. And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: 8n [v · d · e]
Space Spherical Compact hyperbolic Paracompact
Tiling H2-8-3-dual.svg H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png H2 tiling 288-4.png H2 tiling 28i-4.png
Config. 8.8 83 84 85 86 87 88 ...8

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.



See also

  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

References

  1. Grünbaum, Branko (2003). "Are Your Polyhedra the Same as My Polyhedra?". Discrete and Computational Geometry 25: 461–488. doi:10.1007/978-3-642-55566-4_21. http://faculty.washington.edu/moishe/branko/Your%20polyhedra,%20my%20polyhedra/grun.pdf. Retrieved 27 April 2023. 
  • 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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. 

External links