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Order-6 hexagonal tiling

Summary

Order-6 hexagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 66
Schläfli symbol {6,6}
Wythoff symbol 6 | 6 2
Coxeter diagram
Symmetry group [6,6], (*662)
Dual self dual
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.

The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the     tiling:

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}
Spherical Euclidean Hyperbolic tilings

{2,6}

{3,6}

{4,6}

{5,6}

{6,6}
{7,6}

{8,6}

...
{∞,6}

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram      , progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings

{6,2}

{6,3}

{6,4}

{6,5}

{6,6}
{6,7}

{6,8}
...
{6,∞}
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
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{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals

V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12 Alternations [1+,6,6](*663) [6+,6](6*3) [6,1+,6](*3232) [6,6+](6*3) [6,6,1+](*663) [(6,6,2+)](2*33) [6,6]+(662)       =                 =                 =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Similar H2 tilings in *3232 symmetry
Coxeter
diagrams

Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image
Dual

References

• 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.