Rhombitetraapeirogonal tiling
Rhombitetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4..4
Schläfli symbolrr{,4} or
Wythoff symbol4 | 2
Coxeter diagram or
Symmetry group[,4], (*42)
DualDeltoidal tetraapeirogonal tiling
PropertiesVertex-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
Symmetry [∞,4]
(*∞42)
[∞,∞,∞] = [∞,1+,4]
(*∞222)
Schläfli symbol rr{∞,4} t0,1,2,3{∞,∞,∞}
Coxeter diagram

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.

*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
Paracompact uniform tilings in [,4] family
{,4} t{,4} r{,4} 2t{,4}=t{4,} 2r{,4}={4,} rr{,4} tr{,4}
Dual figures
V4 V4.. V(4.)2 V8.8. V4 V43. V4.8.
Alternations
[1+,,4]
(*44)
[+,4]
(*2)
[,1+,4]
(*22)
[,4+]
(4*)
[,4,1+]
(*2)
[(,4,2+)]
(2*2)
[,4]+
(42)

=

=
h{,4} s{,4} hr{,4} s{4,} h{4,} hrr{,4} s{,4}
Alternation duals
V(.4)4 V3.(3.)2 V(4..4)2 V3..(3.4)2 V V.44 V3.3.4.3.

See also

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.
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