Rhombitetrahexagonal tiling

From Wikipedia, the free encyclopedia
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

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 Rhombitetrahexagonal tiling
Image Uniform tiling 64-t02.png Uniform tiling 4.4.4.6.png
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.png

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.pngDeltoidal tetrahexagonal til.pngH2chess 246d.png642 symmetry 0a0.png

Related polyhedra and tiling[]

*n42 symmetry mutation of expanded tilings: n.4.4.4
  • v
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 H2-5-4-cantellated.svg 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 		H2-5-4-deltoidal.svg
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
  • v
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.png H2 tiling 246-3.png H2 tiling 246-2.png H2 tiling 246-6.png H2 tiling 246-4.png H2 tiling 246-5.png H2 tiling 246-7.png
{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.png H2chess 246f.png H2chess 246a.png H2chess 246e.png H2chess 246c.png H2chess 246d.png H2checkers 246.png
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.png Uniform tiling 64-h02.png Uniform tiling 64-h1.png Uniform tiling 443-snub2.png Uniform tiling 66-t0.png Uniform tiling 3.4.4.4.4.png Uniform tiling 64-snub.png
h{6,4} s{6,4} s{4,6} h{4,6} sr{6,4}
Uniform tilings in symmetry *3222
  • v
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png 64
Uniform tiling 64-t0.png 		CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 01.png 6.6.4.4
Uniform tiling 6.6.4.4 (green).png 		CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 01.png (3.4.4)2
Uniform tiling 3.4.4.3.4.4.png 		CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png 4.3.4.3.3.3
Uniform tiling 4.3.4.3.3.3.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 10.png 6.6.4.4
Uniform tiling 6.6.4.4.png 		CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png 6.4.4.4
Uniform tiling 4.4.4.6.png 		CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4 (green).png
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png (3.4.4)2
Uniform tiling 64-h1.png 		CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4.png 		CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png 46
Uniform tiling 64-t2.png

See also[]

  • 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.

External links[]

Retrieved from ""