Truncated hexaoctagonal tiling
Truncated hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.12.16
Schläfli symboltr{8,6} or
Wythoff symbol2 8 6 |
Coxeter diagram or
Symmetry group[8,6], (*862)
DualOrder-6-8 kisrhombille tiling
PropertiesVertex-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

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
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

See also

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

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.
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