Order-6 apeirogonal tiling
Order-6 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration∞6
Schläfli symbol{∞,6}
Wythoff symbol6 | ∞ 2
Coxeter diagram
Symmetry group[∞,6], (*∞62)
DualInfinite-order hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive edge-transitive

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

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 six faces per vertex, starting with the triangular tiling, with Schläfli symbol {n,6}, and Coxeter diagram , with n progressing to infinity.

Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings

{2,6}
{3,6}
{4,6}
{5,6}
{6,6}
{7,6}
{8,6}
 ...
{∞,6}

See also

Wikimedia Commons has media related to Order-6 apeirogonal tiling.

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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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