# Rhombitetrahexagonal tiling

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.

Rhombitetrahexagonal tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 4.4.6.4 |

Schläfli symbol | rr{6,4} or |

Wythoff symbol | 4 | 6 2 |

Coxeter diagram | |

Symmetry group | [6,4], (*642) |

Dual | Deltoidal tetrahexagonal tiling |

Properties | Vertex-transitive |

## 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).

Name | Rhombitetrahexagonal tiling | |
---|---|---|

Image | ||

Symmetry | [6,4] (*642) |
[6,1^{+},4] = [∞,3,∞](*3222) |

Schläfli symbol | rr{6,4} | t_{0,1,2,3}{∞,3,∞} |

Coxeter diagram |

There are 3 lower symmetry forms seen by including edge-colorings: ^{+}] (4*3) symmetry. ^{+},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 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

[6,4], (*632) |
[6,4 ^{+}], (4*3) | ||||||||||

[6 ^{+},4], (6*2) |
[6 ^{+},4^{+}], (32×) |

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

*n42 symmetry mutation of expanded tilings: n.4.4.4 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry [n,4], (* n42) |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | |||||||

*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4] |
*∞42 [∞,4] | |||||

Expanded figures |
|||||||||||

Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4 | ||||

Rhombic figures config. |
V3.4.4.4 |
V4.4.4.4 |
V5.4.4.4 |
V6.4.4.4 |
V7.4.4.4 |
V8.4.4.4 |
V∞.4.4.4 |

Uniform tetrahexagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [6,4], (*642)(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||

= = = |
= |
= = = |
= |
= = = |
= |
||||||

{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||

Uniform duals | |||||||||||

V6^{4} |
V4.12.12 | V(4.6)^{2} |
V6.8.8 | V4^{6} |
V4.4.4.6 | V4.8.12 | |||||

Alternations | |||||||||||

[1^{+},6,4](*443) |
[6^{+},4](6*2) |
[6,1^{+},4](*3222) |
[6,4^{+}](4*3) |
[6,4,1^{+}](*662) |
[(6,4,2^{+})](2*32) |
[6,4]^{+}(642) | |||||

= |
= |
= |
= |
= |
= |
||||||

h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |

Uniform tilings in symmetry *3222 | ||||
---|---|---|---|---|

^{4} |
^{2} |
|||

^{2} |
^{6} |

## See also

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

- Square tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes

## References

- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations) - "Chapter 10: Regular honeycombs in hyperbolic space".
*The Beauty of Geometry: Twelve Essays*. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.