Rhombitetraapeirogonal tiling

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Rhombitetraapeirogonal tiling
Rhombitetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.4.∞.4
Schläfli symbol rr{∞,4} or
Wythoff symbol 4 | ∞ 2
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png or CDel node.pngCDel split1-i4.pngCDel nodes 11.png
Symmetry group [∞,4], (*∞42)
Dual Deltoidal tetraapeirogonal tiling
Properties Vertex-transitive

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

Constructions[]

There are two uniform constructions of this tiling, one from [∞,4] or (*∞42) symmetry, and secondly removing the mirror middle, [∞,1+,4], gives a rectangular fundamental domain [∞,∞,∞], (*∞222).

Two uniform constructions of 4.4.4.∞
Name Rhombitetrahexagonal tiling
Image H2 tiling 24i-5.png Uniform tiling i222-t0123.png
Symmetry [∞,4]
(*∞42)
CDel node c1.pngCDel infin.pngCDel node c3.pngCDel 4.pngCDel node c2.png 		[∞,∞,∞] = [∞,1+,4]
(*∞222)
CDel nodeab c1-2.pngCDel ia2b-cross.pngCDel nodeab c1-2.png
Schläfli symbol rr{∞,4} t0,1,2,3{∞,∞,∞}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel nodes 11.pngCDel ia2b-cross.pngCDel nodes 11.png

Symmetry[]

The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of (*∞222) orbifold symmetry. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.

H2chess 24id.pngDeltoidal tetraapeirogonal tiling.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
Paracompact uniform tilings in [∞,4] family
  • v
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 24i-1.png H2 tiling 24i-3.png H2 tiling 24i-2.png H2 tiling 24i-6.png H2 tiling 24i-4.png H2 tiling 24i-5.png H2 tiling 24i-7.png
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 24ib.png H2chess 24if.png H2chess 24ia.png H2chess 24ie.png H2chess 24ic.png H2chess 24id.png H2checkers 24i.png
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞) 		[∞+,4]
(∞*2) 		[∞,1+,4]
(*2∞2∞) 		[∞,4+]
(4*∞) 		[∞,4,1+]
(*∞∞2) 		[(∞,4,2+)]
(2*2∞) 		[∞,4]+
(∞42)
CDel node h1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png 		CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-ii.pngCDel nodes 10lu.png 		CDel node h.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
H2 tiling 44i-1.png Uniform tiling i42-h01.png H2 tiling 2ii-1.png Uniform tiling i42-snub.png
Alternation duals
CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png
H2chess 44ib.png H2 tiling 2ii-4.png
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞

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[]

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