Order-6 apeirogonal tiling

Order-6 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	∞⁶
Schläfli symbol	{∞,6}
Wythoff symbol	6 \| ∞ 2
Coxeter diagram
Symmetry group	[∞,6], (*∞62)
Dual	Infinite-order hexagonal tiling
Properties	Vertex-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.

Related polyhedra and tiling

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} v t e
Spherical	Euclidean	Hyperbolic tilings
{2,6}	{3,6}	{4,6}	{5,6}	{6,6}	{7,6}	{8,6}	...	{∞,6}

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.