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.

See also

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

References

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