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 $r\left\{\begin{array}{c}\mathrm{\infty }\\ 3\end{array}\right\}$ s2{3,∞}
Wythoff symbol	3 | ∞ 2
Coxeter diagram	File:CDel infin.pngFile:CDel 3.png or File:CDel split1-i3.pngFile:CDel nodes 11.png File:CDel infin.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel 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}.

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 File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel infin.png, 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, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel infin.png.

Related polyhedra and tiling

Template:Order i-3 tiling table

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. Template:Expanded table

See also

• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Search this book on . (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. Search this book on

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.

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