Kisrhombille tiling {{Infobox face-uniform tiling | name=Kisrhombille tiling| Image_File=1-uniform 3 dual.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=V4.6.12| 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 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 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. {{Clear}} SymmetryThe 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}} 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.{{citation needed|date=January 2013}} {{Clear}} 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.) {{Hexagonal tiling table}} Symmetry mutationsThis 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}} See also {{Commons category|Uniform tiling 4-6-12}}- Tilings of regular polygons
- List of uniform tilings
Notes 1. ^1 Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table
2. ^{{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|ref=harv}}
3. ^{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/uniftil.htm |title=Archived copy |accessdate=2006-09-09 |deadurl=yes |archiveurl=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archivedate=2006-09-09 |df= }}
4. ^Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D
References - {{The Geometrical Foundation of Natural Structure (book)|page=41}}
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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
External links- {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
- {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
- {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|x3x6x - othat - O9}}
{{Tessellation}} 4 : Euclidean tilings|Isogonal tilings|Semiregular tilings|Truncated tilings |