# Truncated infinite-order triangular tiling

Infinite-order truncated triangular tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex figure ∞.6.6
Schläfli symbol t{3,∞}
Wythoff symbol 2 ∞ | 3
Coxeter diagram
Symmetry group [∞,3], (*∞32)
Dual apeirokis apeirogonal tiling
Properties Vertex-transitive

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.

## Symmetry

Truncated infinite-order triangular tiling with mirror lines

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

Small index subgroups of [(∞,3,3)], (*∞33)
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)
[(∞,3,3)]

(*∞33)
[(∞,3,3)]+

(∞33)

## Related polyhedra and tiling

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

Dimensional family of truncated spherical polyhedra and tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact hyperb. Parac. Noncompact hyperbolic
*232
[2,3]
D3h
*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] [3i,3]
Figures
Schläfli t{3,2} t{3,3} t{3,4} t{3,5} t{3,6} t{3,7} t{3,8} t{3,∞} t{3,12i} t{3,9i} t{3,6i} t{3,3i}
Coxeter
Uniform dual figures
n-kis
figures
Config.

V2.6.6

V3.6.6

V4.6.6

V5.6.6

V6.6.6

V7.6.6

V8.6.6

V∞.6.6
V12i.6.6 V9i.6.6 V6i.6.6 V3i.6.6
Coxeter
Paracompact 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.∞
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.∞