Английская Википедия:Infinite-order triangular tiling

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Шаблон:Uniform hyperbolic tiles db

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

Symmetry

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

Файл:Infinite-order triangular tiling.svg
Alternated colored tiling
Файл:Iii symmetry mirrors.png
*∞∞∞ symmetry
Файл:Apolleangasket symmetry.png
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}. Шаблон:Triangular regular tiling

Шаблон:Order i-3 tiling table Шаблон: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:

Файл:Ideal-triangle hyperbolic tiling.svg

See also

Шаблон:Commons category

References

External links

Шаблон:Tessellation