Truncated trihexagonal tiling  

Type  Semiregular tiling 
Vertex configuration  4.6.12 
Schläfli symbol  tr{6,3} or 
Wythoff symbol  2 6 3  
Coxeter diagram  
Symmetry  p6m, [6,3], (*632) 
Rotation symmetry  p6, [6,3]^{+}, (632) 
Bowers acronym  Othat 
Dual  Kisrhombille tiling 
Properties  Vertextransitive 
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.
The name truncated trihexagonal tiling is analogous to truncated cuboctahedron and truncated icosidodecahedron, and misleading in the same way. An actual truncation of the trihexagonal tiling has rectangles instead of squares, and its hexagonal and dodecagonal faces can not both be regular. Alternate interchangeable names are:


There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2uniform coloring has two colors of hexagons. 3uniform colorings can have 3 colors of dodecagons or 3 colors of squares.
1uniform  2uniform  3uniform  

Coloring  
Symmetry  p6m, [6,3], (*632)  p3m1, [3^{[3]}], (*333) 
The truncated trihexagonal tiling has three related 2uniform tilings, one being a 2uniform 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.^{[2]}^{[3]}
Semiregular  Dissections  Semiregular  2uniform  3uniform  



Dual  Insets  
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 (kissing number).^{[4]}
Kisrhombille tiling  

Type  Dual semiregular tiling 
Faces  306090 triangle 
Coxeter diagram  
Symmetry group  p6m, [6,3], (*632) 
Rotation group  p6, [6,3]^{+}, (632) 
Dual polyhedron  truncated trihexagonal tiling 
Face configuration  V4.6.12 
Properties  facetransitive 
The kisrhombille tiling or 36 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 306090 triangles with 4, 6, and 12 triangles meeting at each vertex.
Subdividing the faces of these tilings creates the kisrhombille tiling. (Compare the disdyakis hexa, dodeca and triacontahedron, three Catalan solids similar to this tiling.)
Conway calls it a kisrhombille^{[1]} for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 36 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 37 kisrhombille.
It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.)
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.
The kisrhombille tiling triangles represent the fundamental domains of p6m, [6,3] (*632 orbifold notation) wallpaper group symmetry. There are a number of small index subgroups constructed from [6,3] by mirror removal and alternation. [1^{+},6,3] creates *333 symmetry, shown as red mirror lines. [6,3^{+}] creates 3*3 symmetry. [6,3]^{+} is the rotational subgroup. The commutator subgroup is [1^{+},6,3^{+}], which is 333 symmetry. A larger index 6 subgroup constructed as [6,3*], also becomes (*333), shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.
Small index subgroups [6,3] (*632)  

Index  1  2  3  6  
Diagram  
Intl (orb.) Coxeter 
p6m (*632) [6,3] = = 
p3m1 (*333) [1^{+},6,3] = = 
p31m (3*3) [6,3^{+}] = 
cmm (2*22)  pmm (*2222)  p3m1 (*333) [6,3*] = =  
Direct subgroups  
Index  2  4  6  12  
Diagram  
Intl (orb.) Coxeter 
p6 (632) [6,3]^{+} = = 
p3 (333) [1^{+},6,3^{+}] = = 
p2 (2222)  p2 (2222)  p3 (333) [1^{+},6,3*] = = 
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular 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. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Uniform hexagonal/triangular tilings  

Symmetry: [6,3], (*632)  [6,3]^{+} (632) 
[6,3^{+}] (3*3)  
{6,3}  t{6,3}  r{6,3}  t{3,6}  {3,6}  rr{6,3}  tr{6,3}  sr{6,3}  s{3,6}  
6^{3}  3.12^{2}  (3.6)^{2}  6.6.6  3^{6}  3.4.6.4  4.6.12  3.3.3.3.6  3.3.3.3.3.3  
Uniform duals  
V6^{3}  V3.12^{2}  V(3.6)^{2}  V6^{3}  V3^{6}  V3.4.6.4  V.4.6.12  V3^{4}.6  V3^{6} 
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and CoxeterDynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n  

Sym. *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 
[12i,3] 
[9i,3] 
[6i,3] 
[3i,3]  
Figures  
Config.  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞  4.6.24i  4.6.18i  4.6.12i  4.6.6i 
Duals  
Config.  V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞  V4.6.24i  V4.6.18i  V4.6.12i  V4.6.6i 