Rhombitetrahexagonal tiling


In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetrahexagonal tiling, r, 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 or symmetry, and secondly removing the mirror middle, , gives a rectangular fundamental domain ,.
NameRhombitetrahexagonal tiling
Image
Symmetry

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Schläfli symbolrrt0,1,2,3
Coxeter diagram =

There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with symmetry. sees the yellow squares as rectangles, with two color edges, with symmetry. A final quarter symmetry combines these colorings, with 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 642 symmetry|, triangular symmetry with one mirror removed, constructed as ,. Removing half of the blue mirrors doubles the domain again into *3322 symmetry.

Related polyhedra and tiling