Deltoidal trihexagonal tiling

Deltoidal trihexagonal tiling
Type	Dual semiregular tiling
Faces	kite
Face configuration	V3.4.6.4
Symmetry group	p6m, [6,4], (*632)
Dual	Rhombitrihexagonal tiling
Properties	face-transitive

In geometry, the deltoidal trihexagonal tiling is a dual of the semiregular tiling.

Conway calls it a tetrille.^[1]

The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling.

Dual tiling

The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.^[2] Its faces are deltoids or kites.

Topological relations

This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.

This tiling is topologically related to three catalan solids, with face configurations 3.4.n.4, and continues into tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

V3.4.3.4 (*332) and (*432)

V3.4.4.4 (*432)

V3.4.5.4 (*532)

V3.4.6.4 (*632)

V3.4.7.4 (*732)

See also

• Tilings of regular polygons
• List of uniform planar tilings

Notes

1. ^ Conway, 2008, p288 table
2. ^ Weisstein, Eric W., "Dual tessellation" from MathWorld. (See comparative overlay of this tiling and its dual)

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p40
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.  (Page 476, Tilings by polygons, #41 of 56 polygonal isohedral types by quadrangles)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings)