Rhombitriapeirogonal tiling


Rhombitriapeirogonal tiling
Rhombitriapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.4.∞.4
Schläfli symbol rr{∞,3} or
Wythoff symbol 3 | ∞ 2
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png or CDel node.pngCDel split1-i3.pngCDel nodes 11.png
CDel node 1.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
Symmetry group [∞,3], (*∞32)
[∞,3+], (3*∞)
Dual Deltoidal triapeirogonal tiling
Properties Vertex-transitive

In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.


This tiling has [∞,3], (*∞32) symmetry. There is only one uniform coloring.

Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*∞) orbifold notation. The apeireogons can be considered as truncated, t{∞} with two types of edges. It has Coxeter diagram      , Schläfli symbol s2{3,∞}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an infinite-order triangular tiling results, constructed as a snub triapeirotrigonal tiling,      .

Related polyhedra and tilingEdit

Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.∞

Symmetry mutationsEdit

This hyperbolic tiling is topologically related as a part of sequence of uniform cantellated polyhedra with vertex configurations (3.4.n.4), and [n,3] Coxeter group symmetry.

*n42 symmetry mutation of expanded tilings: 3.4.n.4
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
Config. 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4 See alsoEdit


  • 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 linksEdit