Truncated trihexagonal tiling

Truncated trihexagonal tiling
Truncated trihexagonal tiling
Type	Semiregular tiling
Vertex configuration	File:Tiling great rhombi 3-6 vertfig.svg 4.6.12
Schläfli symbol	tr{6,3} or $t\left\{\begin{array}{c}6\\ 3\end{array}\right\}$
Wythoff symbol	2 6 3 |
Coxeter diagram	Template:CDD
Symmetry	p6m, [6,3], (*632)
Rotation symmetry	p6, [6,3]+, (632)
Bowers acronym	Othat
Dual	Kisrhombille tiling
Properties	Vertex-transitive

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}.

File:Rhombic truncated trihexagonal tiling.svg

An equilateral variation with rhombi instead of squares, and isotoxal hexagons instead of regular

Names

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: Great rhombitrihexagonal tiling Rhombitruncated trihexagonal tiling Omnitruncated hexagonal tiling, omnitruncated triangular tiling Conway calls it a truncated hexadeltille.

File:Tiling 3-6 simple.svg File:Tiling 3-6, truncated.svg Trihexagonal tiling and its truncation

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.

	1-uniform	2-uniform	3-uniform
Coloring	File:Uniform polyhedron-63-t012.png	File:Uniform polyhedron-63-t012b.png	File:Uniform polyhedron-63-t012c.png	File:Uniform polyhedron-63-t012d.png
Symmetry	p6m, [6,3], (*632)	p3m1, [3[3]], (*333)

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.

Semiregular	Dissections	Semiregular	2-uniform		3-uniform
File:1-uniform n3.svg	File:Regular hexagon.svgFile:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg File:Hexagonal cupola flat.svgFile:Regular dodecagon.svg	File:1-uniform 6b.png	File:2-uniform 5b.png	File:2-uniform 13b.png	File:3-uniform 6b.png
Dual	Insets
File:1-Uniform 3.png	File:Inset Variations of Dual Uniform Tiling.svg	File:3 Inset to D.gif	File:3 Inset to rD.gif	File:3 Inset to SH.gif	File:3 Inset to 3SH.gif

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).

File:1-uniform-3-circlepack.svg

Kisrhombille tiling

Kisrhombille tiling
Type	Dual semiregular tiling
Coxeter diagram	Template:CDD
Wallpaper group	p6m, [6,3], (*632)
Rotation group	p6, [6,3]+, (632)
Dual	truncated trihexagonal tiling
Face configuration	V4.6.12File:Tiling great rhombi 3-6 dual face.svg
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
  3-6 deltoidal
• rhombille
  rhombille
• hexagonal
  hexagonal
• hexagonal (as exploded triangular)
  hexagonal
  (as exploded triangular)
• triangular
  triangular
• triangular (as hexakis hexagonal)
  triangular
  (as hexakis hexagonal)
• triakis triangular
  triakis triangular

File:Kisrhombille under truncated trihexagonal.svg

File:Kisrhombille under floret pentagonal left (light faces).svg

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

Construction from rhombille tiling

Conway 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.

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	File:632 symmetry lines.png	File:632 symmetry lines-b.png	File:632 symmetry lines-c.png	File:632 symmetry lines-lambda.png	File:632 symmetry lines-lambda-2.png	File:632 symmetry lines-a.png
Intl (orb.) Coxeter	p6m (*632) [6,3] = Template:CDD = Template:CDD	p3m1 (*333) [1+,6,3] = Template:CDD = Template:CDD	p31m (3*3) [6,3+] = Template:CDD	cmm (2*22)	pmm (*2222)	p3m1 (*333) [6,3*] = Template:CDD = Template:CDD
Direct subgroups
Index	2	4		6	12
Diagram	File:632 symmetry alternated.png	File:632 symmetry lines-b2.png		File:632 symmetry lines-delta.png	File:632 symmetry lines-delta-2.png	File:632 symmetry lines-a2.png
Intl (orb.) Coxeter	p6 (632) [6,3]+ = Template:CDD = Template:CDD	p3 (333) [1+,6,3+] = Template:CDD = Template:CDD		p2 (2222)	p2 (2222)	p3 (333) [1+,6,3*] = Template:CDD = Template:CDD

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.)

Symmetry mutations

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram Template:CDD. 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.

See also

• Tilings of regular polygons
• List of uniform tilings

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn [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, Template:Isbn, pp. 50–56

External links

• Klitzing, Richard. "2D Euclidean tilings x3x6x - othat - O9".