Order-6 pentagonal tiling

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

Order-6 pentagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration56
Schläfli symbol{5,6}
Wythoff symbol6 | 5 2
Coxeter diagram
Symmetry group[6,5], (*652)
DualOrder-5 hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

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).


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 Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.


    • 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.

    See also

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