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 '''Infiniteinfinite-order trianglartriangular 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 ==

== Related polyhedra and tiling ==

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

This tiling is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]] {3,p}.

|- align=center

{| class="wikitable"

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

|[[Image:Uniform polyhedron-33-t2.png|100px]]<BR>[[Tetrahedron|{3,3}]]

|[[ImageFile:UniformIii polyhedron-43-t2symmetry mirrors.png|100px150px]]<BR>[[Octahedron|{3,4}]] *∞∞∞ symmetry

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

|[[Image:Uniform polyhedron-53-t2.png|100px]]<BR>[[Icosahedron|{3,5}]]

|[[Image:Uniform polyhedron-63-t2.png|100px]]<BR>[[Triangular tiling|{3,6}]]

|-

|[[Image:Uniform tiling 73-t2.png|100px]]<BR>'''{3,7}'''

|[[Image:Uniform tiling 83-t2.png|100px]]<BR>[[Order-8 triangular tiling|{3,8}]]

|[[Image:Uniform tiling 39-t0.png|100px]]<BR>{3,9}

|[[File:H2_tiling_23i-4.png|100px]]<BR>(3,∞}

|}

== Related polyhedra and tiling ==

This tiling is topologically related as a 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===

A nonregular infinite-order triangular tiling can be generated by a [[Recursion (computer science)|recursive]] process from a central triangle as shown here:

:[[File:Ideal-triangle hyperbolic tiling.svg|240px]]

==See also==

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

*[[Infinite-order tetrahedral honeycomb]]

*[[List of regular polytopes]]

*[[List of uniform planar tilings]]

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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)

{{reflist}}

{{refbegin}}

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

[[Category:TessellationHyperbolic tilings]]

[[Category:Infinite-order tilings]]

[[Category:Isogonal tilings]]

[[Category:Isohedral tilings]]

[[Category:Regular tilings]]

[[Category:Triangular tilings]]