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{{Infobox face-uniform tiling | | |||
Image_File=Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg | | |||
Type=[[List of uniform tilings|Dual semiregular tiling]]| | |||
Cox={{CDD|node_f1|3|node_f1|6|node_f1}} | | |||
Face_List=[[triangle|30-60-90 triangle]]| | |||
Symmetry_Group=p6m, [6,3], (*632)| | |||
Rotation_Group = p6, [6,3]<sup>+</sup>, (632) | | |||
Face_Type=V4.6.12| | |||
Dual=[[truncated trihexagonal tiling]]| | |||
Property_List=[[face-transitive]]| | |||
}} | |||
In [[geometry]], the '''kisrhombille tiling''' or '''3-6 kisrhombille tiling''' is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree [[right triangle]]s with 4, 6, and 12 triangles meeting at each vertex. | |||
== Construction from rhombille tiling == | |||
[[John Horton Conway|Conway]] calls it a '''kisrhombille'''<ref>John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 [http://www.akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)</ref> for his [[Conway kis operator|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]]. | |||
[[File:Rhombic star tiling.png|150px|thumb|left|The related [[rhombille tiling]] becomes the kisrhombille by subdivding the rhombic faces on it axes into four triangle faces]] | |||
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. | |||
==Dual tiling== | |||
It is the dual tessellation of the [[truncated trihexagonal tiling]] which has one square and one hexagon and one dodecagon at each vertex. | |||
:[[File:P6 dual.png|320px]] | |||
== Related polyhedra and tilings == | |||
The ''kisrhombille tiling'' is a part of a set of uniform dual tilings, corresponding to the dual of the truncated trihexagonal tiling. | |||
{{Hexagonal tiling table}} | |||
It is also topologically related to a polyhedra sequence defined by the [[face configuration]] ''V4.6.2n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any <math>n \ge 7.</math> | |||
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. | |||
Each face on these domains also corresponds to the fundamental domain of a [[symmetry group]] with order 2,3,n mirrors at each triangle face vertex. | |||
{{Omnitruncated table}} | |||
== Practical uses == | |||
The ''kisrhombille tiling'' is a useful starting point for making paper models of [[deltahedron|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.{{fact|date=January 2013}} | |||
== See also == | |||
{{Commonscat|Kisrhombille tiling}} | |||
* [[Tilings of regular polygons]] | |||
* [[List of uniform tilings]] | |||
* [[Percolation threshold]] | |||
== Notes == | |||
{{reflist}} | |||
== References == | |||
* {{cite book|author=[[Branko Grünbaum|Grünbaum, Branko]] ; and Shephard, G. C.| title=Tilings and Patterns| location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1}} (Chapter 2.1: ''Regular and uniform tilings'', p.58-65) | |||
* {{The Geometrical Foundation of Natural Structure (book)}} p41 | |||
[[Category:Tessellation]] | |||
Revision as of 23:04, 31 January 2014
Template:Infobox face-uniform tiling
In geometry, 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.
Dual tiling
It is the dual tessellation of the truncated trihexagonal tiling which has one square and one hexagon and one dodecagon at each vertex.
Related polyhedra and tilings
The kisrhombille tiling is a part of a set of uniform dual tilings, corresponding to the dual of the truncated trihexagonal tiling. Template:Hexagonal tiling table
It is also topologically related to a polyhedra sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
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.Template:Fact
See also
Notes
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References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Chapter 2.1: Regular and uniform tilings, p.58-65) - Template:The Geometrical Foundation of Natural Structure (book) p41