Infinite-order triangular tiling

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.

Infinite-order triangular tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration3
Schläfli symbol{3,}
Wythoff symbol3 2
Coxeter diagram
Symmetry group[,3], (*32)
DualOrder-3 apeirogonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
The {3,3,} honeycomb has {3,} vertex figures.

Symmetry

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . 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.


Alternated colored tiling

*∞∞∞ symmetry

Apollonian gasket with *∞∞∞ symmetry

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

Other infinite-order triangular tilings

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

See also

References

  • 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.
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