Tetraoctagonal tiling

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

Tetraoctagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.8)2
Schläfli symbolr{8,4} or
rr{8,8}
rr(4,4,4)
t0,1,2,3(,4,,4)
Wythoff symbol8 4
Coxeter diagram or
or

Symmetry group[8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(,4,,4)], (*4242)
DualOrder-8-4 quasiregular rhombic tiling
PropertiesVertex-transitive edge-transitive

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
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.

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.
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