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 mirror middle, [6,1+,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
Two uniform constructions of 4.4.4.6
| Name
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Rhombitetrahexagonal tiling
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| Image
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| Symmetry
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[6,4]
(*642)
    
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[6,1+,4] = [∞,3,∞]
(*3222)
 =   
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| Schläfli symbol
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rr{6,4}
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t0,1,2,3{∞,3,∞}
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| Coxeter diagram
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=  
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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.
| Lower symmetry constructions
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[6,4], (*632)
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[6,4+], (4*3)
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[6+,4], (6*2)
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[6+,4+], (32×)
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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.
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