Order-8 hexagonal tiling

Order-8 hexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure68
Schläfli symbol{6,8}
Wythoff symbol8 | 6 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-6 octagonal tiling
PropertiesVertex-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}.

Uniform constructions

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
= =

Symmetry

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 tiling

See also

Wikimedia Commons has media related to Order-8 hexagonal tiling.

References

    External links

    This article is issued from Wikipedia - version of the Saturday, May 07, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.