Conwaycalls it a truncated hexadeltille, constructed as a truncation operation applied to a trihexagonal tiling.
Uniform colorings
There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares.
Related 2-uniform tilings
The truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a central hexagon and surrounding triangles and square, in two different orientations.
The Truncated trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing.
Kisrhombille tiling
The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree right triangles with 4, 6, and 12 triangles meeting at each vertex.
calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille. becomes the kisrhombille by cutting each rhombic face along its diagonals into four triangular faces It can be seen as an equilateralhexagonal tiling with each hexagon divided into 12 triangles from the center point. It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.
Symmetry
The kisrhombille tiling triangles represent the fundamental domains of p6m, wallpaper group symmetry. There are a number of ] by mirror removal and alternation. creates *333 symmetry, shown as red mirror lines. creates 3*3 symmetry. + is the rotational subgroup. The commutator subgroup is , which is 333 symmetry. A larger index 6 subgroup constructed as , also becomes, shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.
Related polyhedra and tilings
There are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct.
Symmetry mutations
This tiling can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
