Trihexagonal tiling

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Trihexagonal tiling
Type	Uniform tiling
Vertex figure	3.6.3.6
Schläfli symbol
Wythoff symbol	2 | 6 3 3 3 | 3
Coxeter-Dynkin
Symmetry	p6m
Dual	Quasiregular rhombic tiling
Properties	Vertex-transitive
3.6.3.6

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}.

There are 3 regular and 8 semiregular tilings in the plane.

There are two distinct uniform colorings of a trihexagonal tiling. (Naming the colors by indices on the 3 hexagons around a vertex (3.6.3.6): 1212, 1232.)

[edit] Related polyhedra and tilings

This tiling is topologically related as a part of sequence of rectified polyhedra with vertex figure (3.n.3.n). In this sequence, the edges project into great circles of a sphere on the polyhedra and infinite lines in the planar tiling.

(3.3.3.3)	(3.4.3.4)	(3.5.3.5)
(3.6.3.6)	(3.7.3.7)	(3.8.3.8)

And 3-colors with even orders: 3.2n.3.2n:

(3.4.3.4)

(3.6.3.6)

(3.8.3.8)

[edit] See also

• Tilings of regular polygons
• List of uniform tilings
• Kagome lattice

[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. p38