Infinite-order triangular tiling: Difference between revisions

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{{Short description|Concept in geometry}}

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

[[File:H3 33inf UHS plane at infinity.png|thumb|The [[Infinite-order tetrahedral honeycomb|{3,3,∞∞}]] honeycomb has {3,∞∞} vertex figures.]]

In [[geometry]], the '''infinite-order triangular tiling''' is a [[regular hyperbolic tiling|regular tiling]] of the [[hyperbolic geometry|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 [[Iii symmetry|*∞∞∞∞∞∞ symmetry]], which can be seen with 3 colors of lines representing 3 mirrors of the construction.

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==References==

* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-StrassStrauss, ''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}}