Order-4 octagonal tiling

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

Order-4 octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration84
Schläfli symbol{8,4}
r{8,8}
Wythoff symbol8 2
Coxeter diagram
or
Symmetry group[8,4], (*842)
[8,8], (*882)
DualOrder-8 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.

Four uniform constructions of 8.8.8.8
Uniform
Coloring
Symmetry [8,4]
(*842)
 [8,8]
(*882)
= 		[(8,4,8)] = [8,8,1+]
(*884)
=

=

[1+,8,8,1+]
(*4444)
=
Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}​1⁄2 r{8,4}​1⁄8 = r{8,8}​1⁄4
Coxeter
diagram		 =

=

= =
=

Symmetry

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.


*444
*4222
*832

The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.


{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
 ...
{∞,4}

See also

References
  • 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.