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Order-8 hexagonal tiling
Order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 68
Schläfli symbol {6,8}
Wythoff symbol 8 | 6 2
Coxeter diagram CDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node 1.png
Symmetry group [8,6], (*862)
Dual Order-6 octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.

Contents

Uniform constructionsEdit

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).

Four uniform constructions of 6.6.6.6.6.6.6.6
Uniform
Coloring
   
Symmetry [6,8]
(*862)
     
[6,8,1+] = [(6,6,4)]
(*664)
      =    
[6,1+,8]
(*4232)
      =     
[6,8*]
(*33333333)
Symbol {6,8} {6,8}​12 r(8,6,8) {6,8}​18
Coxeter
diagram
            =            =     

SymmetryEdit

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.

Related polyhedra and tilingEdit

See alsoEdit

ReferencesEdit

  • 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.

External linksEdit