# Tetraapeirogonal tiling

(Redirected from Tetrapeirogonal tiling)
tetraapeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.∞)2
Schläfli symbol r{∞,4} or ${\displaystyle {\begin{Bmatrix}\infty \\4\end{Bmatrix}}}$
rr{∞,∞} or ${\displaystyle r{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}}$
Wythoff symbol 2 | ∞ 4
∞ | ∞ 2
Coxeter diagram
or
Symmetry group [∞,4], (*∞42)
[∞,∞], (*∞∞2)
Dual Order-4-infinite rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetrapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

## Uniform constructions

There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:

Symmetry (*∞42)
[∞,4]
(*∞33)
[1+,∞,4] = [(∞,4,4)]
(*∞∞2)
[∞,4,1+] = [∞,∞]
(*∞2∞2)
[1+,∞,4,1+]
Coxeter = = =
Schläfli r{∞,4} r{4,∞}12 r{∞,4}12=rr{∞,∞} r{∞,4}14
Coloring
Dual

## Symmetry

The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.