Order-6 hexagonal tiling

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Order-6 hexagonal tiling
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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
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[edit]

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 CDel node 1.pngCDel split1-66.pngCDel branch.png tiling:

H2 tiling 366-2.png

Related polyhedra and tiling[edit]

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 CDel node 1.pngCDel n.pngCDel node.pngCDel 6.pngCDel node.png, progressing to infinity.

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 CDel node 1.pngCDel 6.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

References[edit]

  • 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[edit]

External links[edit]