Order-8 square tiling

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Order-8 square tiling
Order-8 square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 48
Schläfli symbol {4,8}
Wythoff symbol 8 | 4 2
Coxeter diagram CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
Symmetry group [8,4], (*842)
Dual Order-4 octagonal tiling
Properties Vertex-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[edit]

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]}, CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png:

Uniform tiling 444-t1.png H2chess 248b.png

Related polyhedra and tiling[edit]

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

See also[edit]

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.

External links[edit]