Truncated tetraoctagonal tiling

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr.

Dual tiling

Symmetry

There are 15 subgroups constructed from by mirror removal and alternation. 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, is the commutator subgroup of .
A larger subgroup is constructed as, index 8, as, with gyration points removed, becomes or, and another, index 16 as, with gyration points removed as or. And their direct subgroups ⁺, ⁺, subgroup indices 16 and 32 respectively, can be given in orbifold notation as and.

Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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.