Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

Tetrahexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.6)2
Schläfli symbolr{6,4} or
rr{6,6}
r(4,4,3)
t0,1,2,3(,3,,3)
Wythoff symbol2 | 6 4
Coxeter diagram or
or

Symmetry group[6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(,3,,3)], (*3232)
DualOrder-6-4 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 [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).

Four uniform constructions of 4.6.4.6
Uniform
Coloring
Fundamental
Domains
Schläfli r{6,4} r{4,6}12 r{6,4}12 r{6,4}14
Symmetry [6,4]
(*642)
[6,6] = [6,4,1+]
(*662)
[(4,4,3)] = [1+,6,4]
(*443)
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
or
Symbol r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
= = =
or

Symmetry

The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) 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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