Infinite-order triangular tiling
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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.
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 |
Related polyhedra and tiling
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
- Infinite-order tetrahedral honeycomb
- List of regular polytopes
- List of uniform planar tilings
- Tilings of regular polygons
- Triangular tiling
- Uniform tilings in hyperbolic plane
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.
External links
- Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html.
- Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html.
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