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
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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
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).

Two uniform constructions of 4.4.4.6
Name Rhombitetrahexagonal tiling
Image Uniform tiling 64-t02 Uniform tiling 4.4.4.6
Symmetry [6,4]
(*642)
CDel node c1.pngCDel 6.pngCDel node c3.pngCDel 4.pngCDel node c2.png
[6,1+,4] = [∞,3,∞]
(*3222)
CDel node c1.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel branch c1.pngCDel 2a2b-cross.pngCDel nodeab c2.png
Schläfli symbol rr{6,4} t0,1,2,3{∞,3,∞}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node 1.png = CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png

There are 3 lower symmetry forms seen by including edge-colorings: CDel node 1.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png sees the hexagons as truncated triangles, with two color edges, with [6,4+] (4*3) symmetry. CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node 1.png 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 CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png.

Skew polyhedron 4446a
Skew polyhedron 4446a

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.

Hyperbolic domains 3222
Deltoidal tetrahexagonal til
H2chess 246d
642 symmetry 0a0
Hyperbolic domains 3222
Deltoidal tetrahexagonal til
H2chess 246d
642 symmetry 0a0

See also

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.

External links

Order-4 hexagonal tiling honeycomb

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 tiling

In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.

Lower symmetry constructions
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
Rhombitetrahexagonal tiling1
[6,4], (*632)
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Rhombitetrahexagonal tiling2
[6,4+], (4*3)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node 1.png
Rhombitetrahexagonal tiling3
[6+,4], (6*2)
CDel 2.png
Rhombitetrahexagonal tiling4
[6+,4+], (32×)
*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
Uniform tiling 432-t02.png Uniform tiling 44-t02.png Uniform tiling 54-t02.png Uniform tiling 64-t02.png Uniform tiling 74-t02.png Uniform tiling 84-t02.png H2 tiling 24i-5.png
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.
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Uniform tiling 44-t0.svg
V4.4.4.4
Deltoidal tetrapentagonal tiling.png
V5.4.4.4
Deltoidal tetrahexagonal til.png
V6.4.4.4
Deltoidal tetraheptagonal til.png
V7.4.4.4
Deltoidal tetraoctagonal til.png
V8.4.4.4
Deltoidal tetraapeirogonal tiling.png
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)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-66.pngCDel nodes 11.png
= CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
= CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 246-1 H2 tiling 246-3 H2 tiling 246-2 H2 tiling 246-6 H2 tiling 246-4 H2 tiling 246-5 H2 tiling 246-7
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 246b H2chess 246f H2chess 246a H2chess 246e H2chess 246c H2chess 246d H2checkers 246
V64 V4.12.12 V(4.6)2 V6.8.8 V46 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)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-66.pngCDel branch hh.pngCDel label2.png
CDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-66.pngCDel nodes 10lu.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel 2xa2xb-cross.pngCDel branch hh.pngCDel label2.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 443-t0 Uniform tiling 64-h02 Uniform tiling 64-h1 Uniform tiling 443-snub2 Uniform tiling 66-t0 Uniform tiling 3.4.4.4.4 Uniform tiling 64-snub
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
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png 64
Uniform tiling 64-t0
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 01.png 6.6.4.4
Uniform tiling 6.6.4.4 (green)
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 01.png (3.4.4)2
Uniform tiling 3.4.4.3.4.4
CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png 4.3.4.3.3.3
Uniform tiling 4.3.4.3.3.3
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 10.png 6.6.4.4
Uniform tiling 6.6.4.4
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png 6.4.4.4
Uniform tiling 4.4.4.6
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4 (green)
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png (3.4.4)2
Uniform tiling 64-h1
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4
CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png 46
Uniform tiling 64-t2

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