# Rhombitriapeirogonal tiling

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

Rhombitriapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.4..4
Schläfli symbolrr{,3} or ${\displaystyle r{\begin{Bmatrix}\infty \\3\end{Bmatrix}}}$
s2{3,}
Wythoff symbol3 | 2
Coxeter diagram or
Symmetry group[,3], (*32)
[,3+], (3*)
DualDeltoidal triapeirogonal tiling
PropertiesVertex-transitive

## Symmetry

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, .

### Symmetry mutations

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.

## 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.