Truncated trihexagonal tiling
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}.
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 
Other names
 Great rhombitrihexagonal tiling
 Rhombitruncated trihexagonal tiling
 Omnitruncated hexagonal tiling, omnitruncated triangular tiling
 Conway calls it a truncated hexadeltille, constructed as a truncation operation applied to a trihexagonal tiling (hexadeltille).[1]
Uniform colorings
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) 
Related 2uniform tilings
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  Dissected  2uniform  3uniform 



Dissected  Semiregular  2uniform  
Circle packing
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
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 3060 degree right triangles with 4, 6, and 12 triangles meeting at each vertex.
Construction from rhombille 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.
Symmetry
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*] = 
Practical uses
The kisrhombille tiling is a useful starting point for making paper models of deltahedra, as each of the equilateral triangles can serve as faces, the edges of which adjoin isosceles triangles that can serve as tabs for gluing the model together.
Related polyhedra and tilings
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} 
Symmetry mutations
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and CoxeterDynkin diagram
*n32 symmetry mutations 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 
See also
Wikimedia Commons has media related to Uniform tiling 4612. 
Notes
 Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table
 Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/08981221(89)901569.
 "Archived copy". Archived from the original on 20060909. Retrieved 20060909.CS1 maint: archived copy as title (link)
 Order in Space: A design source book, Keith Critchlow, p.7475, pattern D
References
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 41. ISBN 048623729X.
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205
 Keith Critchlow, Order in Space: A design source book, 1970, p. 6961, Pattern G, Dual p. 7776, pattern 4
 Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 9780866514613, pp. 50–56