Order-6 pentagonal tiling

Order-6 pentagonal tiling
Order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 56
SchlÀfli 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 SchlÀfli 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.

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

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.