Infinite-order triangular tiling: Difference between revisions

Revision as of 17:41, 5 November 2014

Latest revision as of 11:18, 15 March 2025

(6 intermediate revisions by 6 users not shown)

Line 1:

{{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.

{| class=wikitable width=450

|- align=center

|[[File:H2 tiling 33i-4.png|200px]]<BR>Alternated colored tiling

|[[File:IiiInfinite-order symmetrytriangular mirrorstiling.pngsvg|200px150px]]<BR>Alternated *∞∞∞colored symmetrytiling

|[[File:ApolleangasketIii symmetry mirrors.png|200px150px]]<BR>[[Apollonian gasket]] with *∞∞∞ symmetry

|[[File:Apolleangasket symmetry.png|150px]]<BR>[[Apollonian gasket]] with *∞∞∞ symmetry

|}

Line 31 ⟶ 33:

==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}}

Line 45 ⟶ 47:

[[Category:Isohedral tilings]]

[[Category:Regular tilings]]

[[Category:Triangular tilings]]