Trihexagonal tiling
| Trihexagonal tiling | |
|---|---|
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| Type | Semiregular tiling |
| Vertex configuration | 3.6.3.6 (or (3.6)2) |
| Schläfli symbol | t1{6,3} |
| Wythoff symbol | 2 | 6 3 3 3 | 3 |
| Coxeter-Dynkin |       |
| Symmetry | p6m, [6,3], *632 p3m1, [3[3]], *333 |
| Dual | Rhombille tiling |
| Properties | Vertex-transitive Edge-transitive |
 Vertex figure: 3.6.3.6 (or (3.6)2) |
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In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex. It has Schläfli symbol of t1{6,3}; its edges form an infinite arrangement of lines.
Conway calls it a hexadeltille, combining alternate elements from a hexagonal tiling (Hextille) and triangular tiling (deltille).
There are 3 regular and 8 semiregular tilings in the plane.
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[edit] Uniform colorings
There are two distinct uniform colorings of a trihexagonal tiling. (Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232.)
| Coloring |  |
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|---|---|---|
| Wythoff symbol | 2 | 6 3 | 3 3 | 3 |
| Coxeter-Dynkin diagram | |
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[edit] Related polyhedra and tilings
This tiling is topologically part of sequence of rectified polyhedra with vertex figure (3.n.3.n) and (*n32) reflectional symmetry.
 (3.3.3.3) (*332) and (*432) |
 (3.4.3.4) (*432) |
 (3.5.3.5) (*532) |
(3.6.3.6) (*632) |
 (3.7.3.7) (*732) |
 (3.8.3.8) (*832) |
This tiling is also topologically part of sequence of polyhedra and tilings with vertex figure (3.2n.3.2n) and (*n33) reflectional symmetry.
 (3.4.3.4) (*233) |
(3.6.3.6) (*333) |
 (3.8.3.8) (*433) |
A tiling with alternate large and small triangles is topologically identical to the trihexagonal tiling. The hexagons are distorted so 3 vertices are on the mid-edge of the larger triangles. Similarly there are two uniform colorings:
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[edit] See also
[edit] References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
- 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 (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 38. ISBN 0-486-23729-X.
[edit] External links
- Weisstein, Eric W., "Semiregular tessellation" from MathWorld.
- Richard Klitzing, 2D Euclidean tilings, o3x6o - that - O5
