Order-8 square tiling

Order-8 square tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure48
Schläfli symbol{4,8}
Wythoff symbol8 | 4 2
Coxeter diagram
Symmetry group[8,4], (*842)
DualOrder-4 octagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

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 (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as [1+,8,8,1+], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.

This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4[3]}, :

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

See also

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

References

    This article is issued from Wikipedia - version of the 3/9/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.