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_infinityH3 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.

{| class=wikitable width=450

:[[File:H2_tiling_33i-4.png|160px]]

|- align=center

|[[File:Infinite-order triangular tiling.svg|150px]]<BR>Alternated colored tiling

|[[File:Iii symmetry mirrors.png|150px]]<BR> *∞∞∞ symmetry

|[[File:Apolleangasket symmetry.png|150px]]<BR>[[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}.

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{{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 [[Recursion (computer science)|recursive]] process from a central triangle as shown here:

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==See also==

{{CommonscatCommons category|Infinite-order triangular tiling}}

*[[Infinite-order tetrahedral honeycomb]]

*[[List of regular polytopes]]

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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}}

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*{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk}}

[[Category:{{Tessellation]]}}

[[Category:Hyperbolic geometry]]

[[Category:Tessellation]]

[[Category:Hyperbolic geometrytilings]]

[[Category:Infinite-order tilings]]

[[Category:Isogonal tilings]]

[[Category:Isohedral tilings]]

[[Category:Regular tilings]]

[[Category:Triangular tilings]]