Truncated hexaoctagonal tiling

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

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)Index124DiagramCoxeter[8,6] = [1+,8,6] = [8,6,1+] = = [8,1+,6] = [1+,8,6,1+] = [8+,6+]Orbifold*862*664*883*4232*434343×Semidirect subgroupsDiagramCoxeter[8,6+][8+,6][(8,6,2+)][8,1+,6,1+] = = = = [1+,8,1+,6] = = = = Orbifold6*48*32*433*444*33Direct subgroupsIndex248DiagramCoxeter[8,6]+ = [8,6+]+ = [8+,6]+ = [8,1+,6]+ = [8+,6+]+ = [1+,8,1+,6,1+] = = = Orbifold86266488342324343Radical subgroupsIndex12241632DiagramCoxeter[8,6*][8*,6][8,6*]+[8*,6]+Orbifold*444444*3333333344444433333333

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.

Truncated hexaoctagonal tiling

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