Infinite-order triangular tiling

Infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex figure	3^∞
Schläfli symbol	{3,∞}
Wythoff symbol	∞ | 3 2
Coxeter diagram
Symmetry group	[∞,3], (*∞32)
Dual	Order-3 apeirogonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

The {3,3,∞} honeycomb has {3,∞} vertex figures.

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.

Symmetry

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

Alternated colored tiling

*∞∞∞ symmetry

Apollonian gasket with *∞∞∞ symmetry

Related polyhedra and tiling

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

*n32 symmetry mutation of regular tilings: 3ⁿ or {3,n}

Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
3.3	3^3	3^4	3^5	3^6	3^7	3^8	3^∞	3^12i	3^9i	3^6i	3^3i

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.∞

Other infinite-order triangular tilings

A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

See also

• Infinite-order tetrahedral honeycomb
• List of regular polytopes
• List of uniform planar tilings
• Tilings of regular polygons
• Triangular tiling
• 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.