Truncated order-8 triangular tiling

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Truncated order-8 triangular tiling
Truncated order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.6.6
Schläfli symbol t{3,8}
Wythoff symbol 2 8 | 3
4 3 3 |
Coxeter diagram CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
Symmetry group [8,3], (*832)
[(4,3,3)], (*433)
Dual Octakis octagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex, it has Schläfli symbol of t{3,8}.

Uniform colors[edit]

H2 tiling 334-7.png
The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of hexagons
Uniform dual tiling 433-t012.png
Dual tiling

Symmetry[edit]

The dual of this tiling represents the fundamental domains of *443 symmetry, it only has one subgroup 443, replacing mirrors with gyration points.

This symmetry can be doubled to 832 symmetry by adding a bisecting mirror to the fundamental domain.

Small index subgroups of [(4,3,3)], (*433)
Type Reflectional Rotational
Index 1 2
Diagram 433 symmetry 000.png 433 symmetry aaa.png
Coxeter
(orbifold)
[(4,3,3)] = CDel node c1.pngCDel split1.pngCDel branch c1.pngCDel label4.png
(*433)
[(4,3,3)]+ = CDel node h2.pngCDel split1.pngCDel branch h2h2.pngCDel label4.png
(433)

Related tilings[edit]

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

It can also be generated from the (4 3 3) hyperbolic tilings:

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

See also[edit]

References[edit]

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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. LCCN 99035678.

External links[edit]