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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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=
See also
{{Commons category|Infinite-order triangular tiling}}
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}}
External links
- {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
- {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk}}
{{Tessellation}}