infinite-order triangular tiling

{{Short description|Concept in geometry}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|Ui3_2}}

File:H3 33inf UHS plane at infinity.png 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.

Symmetry

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, {{CDD|node_1|split1|branch|labelinfin}}. 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.

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Alternated colored tiling

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*∞∞∞ symmetry

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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}.

{{Triangular regular tiling}}

{{Order i-3 tiling table}}

{{Order_i-3-3_tiling_table}}

=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:

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See also

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)
  • {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}