Order-6 pentagonal tiling

Order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	56
Schläfli symbol	{5,6}
Wythoff symbol	6 | 5 2
Coxeter diagram
Symmetry group	[6,5], (*652)
Dual	Order-5 hexagonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

Uniform coloring

This regular tiling can also be constructed from [(5,5,3)] symmetry alternating two colors of pentagons, represented by t₁(5,5,3).

Symmetry

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , 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}

Uniform hexagonal/pentagonal tilings v t e
Symmetry: [6,5], (*652)							[6,5]+, (652)	[6,5+], (5*3)	[1+,6,5], (*553)
{6,5}	t{6,5}	r{6,5}	2t{6,5}=t{5,6}	2r{6,5}={5,6}	rr{6,5}	tr{6,5}	sr{6,5}	s{5,6}	h{6,5}
Uniform duals
V65	V5.12.12	V5.6.5.6	V6.10.10	V56	V4.5.4.6	V4.10.12	V3.3.5.3.6	V3.3.3.5.3.5	V(3.5)5

[(5,5,3)] reflective symmetry uniform tilings

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.

See also

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