Tetraapeirogonal tiling

tetraapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.)2
Schläfli symbolr{,4} or \begin{Bmatrix} \infin \\ 4 \end{Bmatrix}
rr{,} or r\begin{Bmatrix} \infin \\ \infin \end{Bmatrix}
Wythoff symbol2 | 4
| 2
Coxeter diagram
or
Symmetry group[,4], (*42)
[,], (*2)
DualOrder-4-infinite rhombille tiling
PropertiesVertex-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.

Related polyhedra and tiling

See also

Wikimedia Commons has media related to Uniform tiling 4-i-4-i.

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. 

    External links

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