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 ${\displaystyle t{\begin{Bmatrix}8\\6\end{Bmatrix}}}$
Wythoff symbol	2 8 6 |
Coxeter diagram	or
Symmetry group	, (*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}.

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 (*862) symmetry.

There are six reflective subgroup kaleidoscopic constructed from 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, (4343) is the commutator subgroup of .

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

Small index subgroups of (*862)

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

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 symmetry, and 7 with subsymmetry.

Uniform octagonal/hexagonal tilings v t e
Symmetry: , (*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
(*466)	(8*3)	(*4232)	(6*4)	(*883)	(2*43)	+ (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

Wikimedia Commons has media related to Uniform tiling 4-12-16.

• Tilings of regular polygons
• List of uniform planar tilings

• 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.

• 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