Order-6 hexagonal tiling

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.

Order-6 hexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration66
Schläfli symbol{6,6}
Wythoff symbol6 | 6 2
Coxeter diagram
Symmetry group[6,6], (*662)
Dualself dual
PropertiesVertex-transitive, edge-transitive, face-transitive

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:

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)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{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.

See also
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