Truncated hexaoctagonal tiling

Truncated hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.12.16
Schläfli symbol	tr{8,6} or $t{\begin{Bmatrix}8\\6\end{Bmatrix}}$
Wythoff symbol	2 8 6 \|
Coxeter diagram	or
Symmetry group	[8,6], (*862)
Dual	Order-6-8 kisrhombille tiling
Properties	Vertex-transitive

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

Dual tiling


The dual tiling is called an order-6-8 kisrhombille tiling, made as a complete bisection of the order-6 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,6] (*862) symmetry.

Symmetry

Truncated hexaoctagonal tiling with mirror lines

There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. 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⁺,6,1⁺] (4343) is the commutator subgroup of [8,6].

A radical subgroup is constructed as [8,6*], index 12, as [8,6⁺], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8⁺,6], (8*3) with gyration points removed as (*33333333).

Small index subgroups of [8,6] (*862)
Index	1	2			4
Diagram
Coxeter	[8,6] =	[1⁺,8,6] =	[8,6,1⁺] = =	[8,1⁺,6] =	[1⁺,8,6,1⁺] =	[8⁺,6⁺]
Orbifold	*862	*664	*883	*4232	*4343	43×
Semidirect subgroups
Diagram
Coxeter		[8,6⁺]	[8⁺,6]	[(8,6,2⁺)]	[8,1⁺,6,1⁺] = = = =	[1⁺,8,1⁺,6] = = = =
Orbifold		6*4	8*3	2*43	3*44	4*33
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[8,6]⁺ =	[8,6⁺]⁺ =	[8⁺,6]⁺ =	[8,1⁺,6]⁺ =	[8⁺,6⁺]⁺ = [1⁺,8,1⁺,6,1⁺] = = =
Orbifold	862	664	883	4232	4343
Radical subgroups
Index		12	24	16	32
Diagram
Coxeter		[8,6*]	[8*,6]	[8,6*]⁺	[8*,6]⁺
Orbifold		*444444	*33333333	444444	33333333

Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 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 [8,6] symmetry, and 7 with subsymmetry.

Uniform octagonal/hexagonal tilings v t e
Symmetry: [8,6], (*862)


{8,6}	t{8,6}	r{8,6}	2t{8,6}=t{6,8}	2r{8,6}={6,8}	rr{8,6}	tr{8,6}
Uniform duals


V8⁶	V6.16.16	V(6.8)²	V8.12.12	V6⁸	V4.6.4.8	V4.12.16
Alternations
[1⁺,8,6] (*466)	[8⁺,6] (8*3)	[8,1⁺,6] (*4232)	[8,6⁺] (6*4)	[8,6,1⁺] (*883)	[(8,6,2⁺)] (2*43)	[8,6]⁺ (862)


h{8,6}	s{8,6}	hr{8,6}	s{6,8}	h{6,8}	hrr{8,6}	sr{8,6}
Alternation duals


V(4.6)⁶	V3.3.8.3.8.3	V(3.4.4.4)²	V3.4.3.4.3.6	V(3.8)⁸	V3.4⁵	V3.3.6.3.8

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. LCCN 99035678.

External links