Bisected hexagonal tiling
From Wikipedia, the free encyclopedia
| Bisected hexagonal tiling | |
|---|---|
 |
|
| Type | Semiregular tiling |
| Faces | Right triangle |
| Edges | Infinite |
| Vertices | Infinite |
| Face configuration | V4.6.12 |
| Symmetry group | p6m |
| Dual | Great rhombitrihexagonal tiling |
| Properties | face-transitive |
In geometry, the bisected hexagonal tiling is a tiling of the Euclidean plane. It is 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.)
Conway calls it a kisrhombille for his kis vertex bisector operation applied to his rhombille (Quasiregular rhombic tiling).
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. It is the dual tessellation of the great rhombitrihexagonal tiling which has one square and one hexagon and one dodecagon at each vertex.
Contents |
[edit] Related polyhedra and tilings
It is 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 n.
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.
| n | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| Tiling space | Spherical | |||
| Face configuration V4.6.2n |
V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 |
| Symmetry group (Orbifold) |
D3h (*322) D6h (*622) full sym. |
Td (*332) Oh (*432) full sym. |
Oh (*432) | Ih (*532) |
| Symmetry fundamental domain |
 (Order 12) |
 (Order 24) |
 (Order 48) |
 (Order 60) |
| Polyhedron |  |
 |
 |
 |
| Net or tiling |
 |
 |
 |
 |
| n | 6 | 7 |
|---|---|---|
| Tiling space | Euclidean | Hyperbolic |
| Face configuration V4.6.2n |
V4.6.12 | V4.6.14 |
| Symmetry group (orbifold notation) |
(*632) | (*732) |
| Net or tiling |
 |
 |
[edit] Practical uses
The bisected hexagonal 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.
[edit] See also
[edit] References
- Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65)
- Williams, Robert The Geometrical Foundation of Natural Structure: A Source Book of Design New York: Dover, 1979. p41
 |
This geometry-related article is a stub. You can help Wikipedia by expanding it. |
