Truncated trihexagonal tiling#Related 2-uniform tilings

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.

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.{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9| doi-access=free }}{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/uniftil.htm |title=Uniform Tilings |access-date=2006-09-09 |url-status=dead |archive-url=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archive-date=2006-09-09 }}

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).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D

:200px

{{Infobox face-uniform tiling |

name=Kisrhombille tiling|

Image_File=Tiling great rhombi 3-6 dual simple.svg|

Type=Dual semiregular tiling|

Cox={{CDD|node_f1|3|node_f1|6|node_f1}} |

Face_List=30-60-90 triangle|

Symmetry_Group=p6m, [6,3], (*632)|

Rotation_Group = p6, [6,3]⁺, (632) |

Face_Type=V4.6.1260px|

Dual=truncated trihexagonal tiling|

Property_List=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.)

File:Kisrhombille in deltoidal.svg|3-6 deltoidal

File:Kisrhombille in rhombille (blue).svg|rhombille

File:Kisrhombille in hexagonal (red).svg|hexagonal

File:Kisrhombille in triangular exploded to hexagonal (yellow).svg|hexagonal
(as exploded triangular)

File:Kisrhombille in triangular (yellow).svg|triangular

File:Kisrhombille in hexakis hexagonal.svg|triangular
(as hexakis hexagonal)

File:Kisrhombille in triakis triangular.svg|triakis triangular

{{multiple image

| align = left | width = 150

| image1 = Kisrhombille under truncated trihexagonal.svg

| image2 = Kisrhombille under floret pentagonal left (light faces).svg

| footer = 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 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.

{{632 symmetry table}}

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.)

{{Hexagonal tiling table}}

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram {{CDD|node_1|p|node_1|3|node_1}}. 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.

{{Omnitruncated table}}

{{Commons category|Uniform tiling 4-6-12 (truncated trihexagonal tiling)}}

• Tilings of regular polygons
• List of uniform tilings

{{reflist}}

• {{The Geometrical Foundation of Natural Structure (book)|page=41}}
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]
• 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

• {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
• {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
• {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|x3x6x - othat - O9}}

{{Tessellation}}

Category:Euclidean tilings

Category:Isogonal tilings

Category:Semiregular tilings

Category:Truncated tilings