Order-6 apeirogonal tiling

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

Order-6 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration6
Schläfli symbol{,6}
Wythoff symbol 2
Coxeter diagram
Symmetry group[,6], (*62)
DualInfinite-order hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive edge-transitive

Symmetry

The dual to this tiling represents the fundamental domains of [∞,6*] symmetry, orbifold notation *∞∞∞∞∞∞ symmetry, a hexagonal domain with five ideal vertices.

The order-6 apeirogonal tiling can be uniformly colored with 6 colored apeirogons around each vertex, and coxeter diagram: , except ultraparallel branches on the diagonals.

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,6}, and Coxeter diagram , with n progressing to infinity.

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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