Truncated tetraoctagonal tiling

Truncated tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.8.16
Schläfli symbol	tr{8,4} or $t\left\{\begin{array}{c}8\\ 4\end{array}\right\}$
Wythoff symbol	2 8 4 |
Coxeter diagram	or
Symmetry group	[8,4], (*842)
Dual	Order-4-8 kisrhombille tiling
Properties	Vertex-transitive

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

Dual tiling

File:H2checkers 248.png	File:Hyperbolic domains 842.png
The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry.

Symmetry

File:Truncated tetraoctagonal tiling with mirrors.png

Truncated tetraoctagonal tiling with *842, File:CDel node c2.pngFile:CDel node c3.pngFile:CDel node c1.png, mirror lines

There are 15 subgroups constructed from [8,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1⁺,8,1⁺,4,1⁺] (4242) is the commutator subgroup of [8,4].

A larger subgroup is constructed as [8,4*], index 8, as [8,4⁺], (4*4) with gyration points removed, becomes (*4444) or (*4⁴), and another [8*,4], index 16 as [8⁺,4], (8*2) with gyration points removed as (*22222222) or (*2⁸). And their direct subgroups [8,4*]⁺, [8*,4]⁺, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).

Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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 7 forms with full [8,4] symmetry, and 7 with subsymmetry.

Template:Order 8-4 tiling table

Template:Omnitruncated4 table Template:Omnitruncated symmetric table

See also

• Tilings of regular polygons
• List of uniform planar tilings

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Search this book on . (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. Search this book on

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

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