Infinite-order apeirogonal tiling

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

Infinite-order apeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration
Schläfli symbol{,}
Wythoff symbol 2
|
Coxeter diagram
Symmetry group[,], (*2)
[(,,)], (*)
Dualself-dual
PropertiesVertex-transitive, edge-transitive, face-transitive

Symmetry

This tiling represents the fundamental domains of *∞ symmetry.

Uniform colorings

This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.

Domains 0 1 2

symmetry:
[(∞,∞,∞)]  

t0{(∞,∞,∞)}

t1{(∞,∞,∞)}

t2{(∞,∞,∞)}

The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *22 fundamental domain.

a{,} or =

See also

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.
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