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 | |
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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 |
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 | |
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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: 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 .
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.
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, 8 hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.
Rhombitetraapeirogonal tilingIn geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.
Lower symmetry constructions | |||||||||||
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[6,4], (*632) |
[6,4^{+}], (4*3) | ||||||||||
[6^{+},4], (6*2) |
[6^{+},4^{+}], (32×) |
*n42 symmetry mutation of expanded tilings: n.4.4.4 | |||||||||||
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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 |
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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 | |||||||||||
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Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
= = = |
= |
= = = |
= |
= = = |
= |
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{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) | |||||
= |
= |
= |
= |
= |
= |
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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 | ||||
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6^{4} |
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 |
4^{6} |
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