Tetraoctagonal tiling

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Tetraoctagonal tiling
Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.8)2
Schläfli symbol r{8,4} or
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol 2 | 8 4
Coxeter diagram CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel split1-84.pngCDel nodes.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png or CDel node.pngCDel split1-88.pngCDel nodes 11.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel 4a4b.pngCDel branch 11.pngCDel labelinfin.png
Symmetry group [8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(∞,4,∞,4)], (*4242)
Dual Order-8-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

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

Constructions[edit]

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)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8] = [8,4,1+]
(*882)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel nodeab c2.png
[(4,4,4)] = [1+,8,4]
(*444)
CDel node h0.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png = CDel label4.pngCDel branch c2.pngCDel split2-44.pngCDel node c3.png
[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
CDel node h0.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch c2.pngCDel 4a4b.pngCDel branch c2.pngCDel labelinfin.png or CDel nodeab c2.pngCDel 4a4b-cross.pngCDel nodeab c2.png
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 CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-88.pngCDel nodes 11.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch 11.pngCDel 4a4b.pngCDel branch 11.pngCDel labelinfin.png or CDel nodes 11.pngCDel 4a4b-cross.pngCDel nodes 11.png

Symmetry[edit]

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

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]