Infinite-order triangular tiling

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Infinite-order triangular tiling
Infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3
Schläfli symbol {3,∞}
Wythoff symbol ∞ | 3 2
Coxeter diagram CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
Dual Order-3 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive
The {3,3,∞} honeycomb has {3,∞} vertex figures.

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.


A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png. The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

Infinite-order triangular tiling.svg
Alternated colored tiling
Iii symmetry mirrors.png
*∞∞∞ symmetry
Apolleangasket symmetry.png
Apollonian gasket with *∞∞∞ symmetry

Related polyhedra and tiling[edit]

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

Other infinite-order triangular tilings[edit]

A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

Ideal-triangle hyperbolic tiling.svg

See also[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]