Truncated tetraoctagonal tiling

Short description: Semiregular tiling in geometry

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

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

Truncated tetraoctagonal tiling with *842, , 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).

Small index subgroups of [8,4] (*842)
Index	1	2			4
Diagram
Coxeter	[8,4] =	[1+,8,4] =	[8,4,1+] = =	[8,1+,4] =	[1+,8,4,1+] =	[8+,4+]
Orbifold	*842	*444	*882	*4222	*4242	42×
Semidirect subgroups
Diagram
Coxeter		[8,4+]	[8+,4]	[(8,4,2+)]	[8,1+,4,1+] = = = =	[1+,8,1+,4] = = = =
Orbifold		4*4	8*2	2*42	2*44	4*22
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[8,4]+ =	[8,4+]+ =	[8+,4]+ =	[8,1+,4]+ =	[8+,4+]+ = [1+,8,1+,4,1+] = = =
Orbifold	842	444	882	4222	4242
Radical subgroups
Index		8	16		32
Diagram
Coxeter		[8,4*] =	[8*,4]	[8,4*]+ =	[8*,4]+
Orbifold		*4444	*22222222	4444	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.

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 (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

• Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html. 
• Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html. 
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch