Infinite-order square tiling explained

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

Uniform colorings

There is a half symmetry form,, seen with alternating colors:

Symmetry

This tiling represents the mirror lines of

• ∞∞∞∞ symmetry

. The dual to this tiling defines the fundamental domains of (*2^∞) orbifold symmetry.

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4ⁿ).

See also

• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes

References

• Book: The Symmetries of Things. 2008. 978-1-56881-220-5. Chapter 19, The Hyperbolic Archimedean Tessellations. John H. Conway. John Horton Conway. Heidi Burgiel. Chaim Goodman-Strauss.
• Book: The Beauty of Geometry: Twelve Essays. 1999. Dover Publications. 99035678. 0-486-40919-8. Chapter 10: Regular honeycombs in hyperbolic space. H. S. M. Coxeter.

External links

• • • Hyperbolic and Spherical Tiling Gallery