Tetraapeirogonal tiling

From Wikipedia, the free encyclopedia
  (Redirected from Tetrapeirogonal tiling)
Jump to: navigation, search
tetraapeirogonal tiling
Tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.∞)2
Schläfli symbol r{∞,4} or
rr{∞,∞} or
Wythoff symbol 2 | ∞ 4
∞ | ∞ 2
Coxeter diagram CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.png or CDel node.pngCDel split1-ii.pngCDel nodes 11.png
Symmetry group [∞,4], (*∞42)
[∞,∞], (*∞∞2)
Dual Order-4-infinite rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetrapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

Uniform constructions[edit]

There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:

Symmetry (*∞42)
[∞,4]
(*∞33)
[1+,∞,4] = [(∞,4,4)]
(*∞∞2)
[∞,4,1+] = [∞,∞]
(*∞2∞2)
[1+,∞,4,1+]
Coxeter CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel labelinfin.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-ii.pngCDel nodes 11.png CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.pngCDel labelinfin.png
Schläfli r{∞,4} r{4,∞}12 r{∞,4}12=rr{∞,∞} r{∞,4}14
Coloring H2 tiling 24i-2.png H2 tiling 2ii-5.png H2 tiling 44i-3.png Uniform tiling verf-i4i4.png
Dual H2chess 24ia.png H2chess 2iid.png H2chess 44if.png H2chess 2iid.png

Symmetry[edit]

The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.

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]