# Order-4 apeirogonal tiling

Order-4 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 4
Schläfli symbol {∞,4}
r{∞,∞}
t(∞,∞,∞)
t0,1,2,3(∞,∞,∞,∞)
Wythoff symbol 4 | ∞ 2
2 | ∞ ∞
∞ ∞ | ∞
Coxeter diagram

Symmetry group [∞,4], (*∞42)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
(*∞∞∞∞)
Dual Infinite-order square tiling
Properties Vertex-transitive, edge-transitive, face-transitive edge-transitive

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

## Symmetry

This tiling represents the mirror lines of *2 symmetry. It dual to this tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.

## Uniform colorings

Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.

1 color 2 color 3 and 2 colors 4, 3 and 2 colors
[∞,4], (*∞42) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) (*∞∞∞∞)
{∞,4} r{∞,∞}
= {∞,4}​12
t0,2(∞,∞,∞)
= r{∞,∞}​12
t0,1,2,3(∞,∞,∞,∞)
= r{∞,∞}​14 = {∞,4}​18

(1111)

(1212)

(1213)

(1112)

(1234)

(1123)

(1122)
= =
=
= =

## Related polyhedra and tiling

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

## 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.