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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 t { 8 6 } {\displaystyle t{\begin{Bmatrix}8\\6\end{Bmatrix}}}
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}.

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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

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)
Index124
Diagram
Coxeter[8,6] = [1+,8,6] = [8,6,1+] = = [8,1+,6] = [1+,8,6,1+] = [8+,6+]
Orbifold*862*664*883*4232*434343×
Semidirect subgroups
Diagram
Coxeter[8,6+][8+,6][(8,6,2+)][8,1+,6,1+] = = = = [1+,8,1+,6] = = = =
Orbifold6*48*32*433*444*33
Direct subgroups
Index248
Diagram
Coxeter[8,6]+ = [8,6+]+ = [8+,6]+ = [8,1+,6]+ = [8+,6+]+ = [1+,8,1+,6,1+] = = =
Orbifold86266488342324343
Radical subgroups
Index12241632
Diagram
Coxeter[8,6*][8*,6][8,6*]+[8*,6]+
Orbifold*444444*3333333344444433333333

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
V86V6.16.16V(6.8)2V8.12.12V68V4.6.4.8V4.12.16
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)6V3.3.8.3.8.3V(3.4.4.4)2V3.4.3.4.3.6V(3.8)8V3.45V3.3.6.3.8

See also

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