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 node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-i3.pngFile:CDel nodes 11.png File:CDel node 1.pngFile: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.pngFile:CDel node 1.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.pngFile:CDel node.png.

Related polyhedra and 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.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Figure	File:Spherical triangular prism.svg	File:Uniform tiling 332-t02.png	File:Uniform tiling 432-t02.png	File:Uniform tiling 532-t02.png	File:Uniform polyhedron-63-t02.png	File:Rhombitriheptagonal tiling.svg	File:H2-8-3-cantellated.svg	File:H2 tiling 23i-5.png	File:H2 tiling 23j12-5.png	File:H2 tiling 23j9-5.png	File:H2 tiling 23j6-5.png
Config.	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4	3.4.12i.4	3.4.9i.4	3.4.6i.4

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 (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

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