# 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}.

Finite Euclidean Compact hyperbolic Paracompact

{3,2}

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,9}
...
(3,∞}
Noncompact hyperbolic 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.∞

### 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: