Infinite-order apeirogonal tiling

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Infinite-order apeirogonal tiling
Infinite-order apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration
Schläfli symbol {∞,∞}
Wythoff symbol ∞ | ∞ 2
∞ ∞ | ∞
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png
Symmetry group [∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
Dual self-dual
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has an infinite number of apeirogons around all its ideal vertices.

Symmetry[edit]

This tiling represents the fundamental domains of *∞ symmetry.

Uniform colorings[edit]

This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.

Domains 0 1 2
Infinite-order triangular tiling.svg
symmetry:
[(∞,∞,∞)]  CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node.png
H2 tiling iii-1.png
t0{(∞,∞,∞)}
CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.png
H2 tiling iii-2.png
t1{(∞,∞,∞)}
CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png
H2 tiling iii-4.png
t2{(∞,∞,∞)}
CDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png

Related polyhedra and tiling[edit]

The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.

Infinite-order apeirogonal tiling and dual.png
a{∞,∞} or CDel node h3.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png = CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png

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]