Triapeirogonal tiling

In geometry, the triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,3}.

Triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(3.∞)^2
Schläfli symbol	r{∞,3} or ${\displaystyle {\begin{Bmatrix}\infty \\3\end{Bmatrix}}}$
Wythoff symbol	2 | ∞ 3
Coxeter diagram	or
Symmetry group	[∞,3], (*∞32)
Dual	Order-3-infinite rhombille tiling
Properties	Vertex-transitive edge-transitive

Uniform colorings

The half-symmetry form, , has two colors of triangles:

Related polyhedra and tiling

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

Quasiregular tilings: (3.n)^2
Sym. *n32 [n,3]	Spherical			Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] p6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Figure
Figure
Vertex	(3.3)^2	(3.4)^2	(3.5)^2	(3.6)^2	(3.7)^2	(3.8)^2	(3.∞)^2	(3.12i)^2	(3.9i)^2	(3.6i)^2
Schläfli	r{3,3}	r{3,4}	r{3,5}	r{3,6}	r{3,7}	r{3,8}	r{3,∞}	r{3,12i}	r{3,9i}	r{3,6i}
Coxeter
Dual uniform figures
Dual conf.	V(3.3)^2	V(3.4)^2	V(3.5)^2	V(3.6)^2	V(3.7)^2	V(3.8)^2	V(3.∞)^2

Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32)							[∞,3]^+ (∞32)	[1^+,∞,3] (*∞33)		[∞,3^+] (3*∞)
		=	=	=				= or	= or	=
{∞,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		V3.3.3.3.3.∞

Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33)							[(∞,3,3)]^+, (∞33)
(∞,∞,3)	t0,1(∞,3,3)	t1(∞,3,3)	t1,2(∞,3,3)	t2(∞,3,3)	t0,2(∞,3,3)	t0,1,2(∞,3,3)	s(∞,3,3)
Dual tilings
V(3.∞)^3	V3.∞.3.∞	V(3.∞)^3	V3.6.∞.6	V(3.3)^∞	V3.6.∞.6	V6.6.∞	V3.3.3.3.3.∞

See also

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

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.

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• http://bendwavy.org/klitzing/incmats/o3xinfino.htm
• Klitzing, Richard. "2D Non-Compact Tilings". o3x∞o