# Order-3 apeirogonal tiling

Order-3 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3
Schläfli symbol {∞,3}
t{∞,∞}
t(∞,∞,∞)
Wythoff symbol 3 | ∞ 2
2 ∞ | ∞
∞ ∞ ∞ |
Coxeter diagram

Symmetry group [∞,3], (*∞32)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
Dual Infinite-order triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex; each apeirogon is inscribed in a horocycle.

The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.

## Images

Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.

## Uniform colorings

Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:

Regular Truncations

{∞,3}

t0,1{∞,∞}

t1,2{∞,∞}

t{∞[3]}
Hyperbolic triangle groups

[∞,3]

[∞,∞]

[(∞,∞,∞)]

### Symmetry

The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors; the symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.

A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞) with gyration points removed, becomes (*∞).

## Related polyhedra and tilings

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