Truncated trihexagonal tiling

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.

	1-uniform	2-uniform	3-uniform
Coloring
Symmetry	p6m, [6,3], (*632)	p3m1, [3[3]], (*333)

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.^[2][3]

Semiregular	Dissections	Semiregular	2-uniform		3-uniform
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	30-60-90 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	face-transitive

The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60-90 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.)

•  
  3-6 deltoidal
•  
  rhombille
•  
  hexagonal

 

 

The kisrhombille tiling under its dual (left) and under the floret pentagonal tiling (right), from which it can be created as a partial truncation.

Conway calls it a kisrhombille^[1] 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.

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 v t e
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}
63	3.122	(3.6)2	6.6.6	36	3.4.6.4	4.6.12	3.3.3.3.6	3.3.3.3.3.3
Uniform duals
V63	V3.122	V(3.6)2	V63	V36	V3.4.6.4	V.4.6.12	V34.6	V36

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin 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 v t e
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

 

Wikimedia Commons has media related to Uniform tiling 4-6-12 (truncated trihexagonal tiling).

• Tilings of regular polygons
• List of uniform tilings

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 41. ISBN 0-486-23729-X.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern G, Dual p. 77-76, pattern 4
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56

• Weisstein, Eric W. "Uniform tessellation". MathWorld.
• Weisstein, Eric W. "Semiregular tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings x3x6x - othat - O9".