In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Four uniform constructions of 4.8.4.8

Name

Tetra-octagonal tiling

Rhombi-octaoctagonal tiling

Image
Uniform_tiling_84-t1.png
Uniform_tiling_88-t02.png
Uniform_tiling_444-t01.png
4242-uniform_tiling-verf4848.png

Symmetry

[8,4]
(*842)

[8,8] = [8,4,1+]
(*882)
=

[(4,4,4)] = [1+,8,4]
(*444)
=

[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
= or

Schläfli

r{8,4}

rr{8,8}
=r{8,4}1/2

r(4,4,4)
=r{4,8}1/2

t0,1,2,3(∞,4,∞,4)
=r{8,4}1/4

Coxeter

=

=

= or

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

Ord84_qreg_rhombic_til.png
H2chess_248e.png

See also

  • Square tiling
  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)