Tetrahexagonal tiling

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Tetrahexagonal tiling
Tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.6)2
Schläfli symbol r{6,4} or
rr{6,6}
r(4,4,3)
t0,1,2,3(∞,3,∞,3)
Wythoff symbol 2 | 6 4
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel split1-64.pngCDel nodes.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png or CDel node.pngCDel split1-66.pngCDel nodes 11.png
CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
Symmetry group [6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(∞,3,∞,3)], (*3232)
Dual Order-6-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

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

Constructions[edit]

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
H2 tiling 246-2.png H2 tiling 266-5.png H2 tiling 344-5.png 3222-uniform tiling-verf4646.png
Fundamental
Domains
642 symmetry 000.png 642 symmetry 00a.png 642 symmetry a00.png 642 symmetry a0b.png
Schläfli r{6,4} r{4,6}​12 r{6,4}​12 r{6,4}​14
Symmetry [6,4]
(*642)
CDel node c3.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node c2.png
[6,6] = [6,4,1+]
(*662)
CDel node c3.pngCDel split1-66.pngCDel nodeab c1.png
[(4,4,3)] = [1+,6,4]
(*443)
CDel branch c1.pngCDel split2-44.pngCDel node c2.png
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
CDel labelinfin.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel labelinfin.png or CDel nodeab c1.pngCDel 3a3b-cross.pngCDel nodeab c1.png
Symbol r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-66.pngCDel nodes 11.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel branch 11.pngCDel split2-44.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png =
CDel labelinfin.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel labelinfin.png or CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png

Symmetry[edit]

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.

Hyperbolic domains 3232.pngOrd64 qreg rhombic til.pngH2chess 246a.pngOrder-6 hexagonal tiling and dual.png

Related polyhedra and tiling[edit]

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]