Rhombitetrahexagonal tiling

Rhombitetrahexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.6.4
Schläfli symbolrr{6,4}
Wythoff symbol4 | 6 2
Coxeter diagram

Symmetry group[6,4], (*642)
DualDeltoidal tetrahexagonal tiling
PropertiesVertex-transitive

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Constructions

There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the miror middle, [6,1+,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).

Two uniform constructions of 4.4.4.6
Name Rhombitetrahexagonal tiling
Image
Symmetry [6,4]
(*642)
 [6,1+,4] = [∞,3,∞]
(*3222)
=
Schläfli symbol rr{6,4} t0,1,2,3{∞,3,∞}
Coxeter diagram =

There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with [6,4+] (4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with [6+,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6+,4+] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.

This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of .

Symmetry

The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1+,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.

Related polyhedra and tiling

Symmetry *3222
64
 6.6.4.4
 (3.4.4)2
 4.3.4.3.3.3
6.6.4.4
 6.4.4.4
 3.4.4.4.4
(3.4.4)2
 3.4.4.4.4
 46

See also

Wikimedia Commons has media related to Uniform tiling 4-4-4-6.

References

External links

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