As the name implies this tiling is constructed by a truncation operation applied to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.
There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as:
Related polyhedra and tilings
A truncated hexagonal tiling can be contracted in one dimension, reducing dodecagons into decagons. Contracting in second direction reduces decagons into octagons. Contracting a third time make the trihexagonal tiling.
Wythoff constructions from hexagonal and triangular tilings
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. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.[1][2]
1-uniform
Dissection
2-uniform dissections
(3.122)
(3.4.6.4) & (33.42)
(3.4.6.4) & (32.4.3.4)
Dual Tilings
O
to DB
to DC
Circle packing
The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.[3] Every circle is in contact with 3 other circles in the packing (kissing number). This is the lowest density packing that can be created from a uniform tiling.
The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.
In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron.[5]
It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.[6]
It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.[7]
Related duals to uniform tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
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