View source for Triangular tiling

{{short description|Regular tiling of the plane}}
{{Uniform tiles db|Reg tiling stat table|Ut}}

In [[geometry]], the '''triangular tiling''' or '''triangular tessellation''' is one of the three [[Euclidean tilings by convex regular polygons#Regular tilings|regular tilings]] of the [[Euclidean plane]], and is the only such tiling where the constituent shapes are not [[parallelogon]]s. Because the internal angle of the [[equilateral triangle]] is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has [[Schläfli symbol]] of {{math|{3,6}.}}

English mathematician [[John Horton Conway|John Conway]] called it a '''deltille''', named from the triangular shape of the Greek letter [[Delta (letter)|delta]] (Δ). The triangular tiling can also be called a '''kishextille''' by a [[Conway kis operator|kis]] operation that adds a center point and triangles to replace the faces of a [[hextille]].

It is one of [[List of regular polytopes#Euclidean tilings|three regular tilings of the plane]]. The other two are the [[square tiling]] and the [[hexagonal tiling]].

== Uniform colorings ==
[[File:Triangular_tiling_4-color.svg|thumb|A 2-uniform triangular tiling, 4 colored triangles, related to the [[geodesic polyhedron]] as {3,6+}<sub>2,0</sub>.]]
There are 9 distinct [[uniform coloring]]s of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.<ref>''[[Tilings and patterns]]'', p.102-107</ref>

There is one class of [[Archimedean coloring]]s, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.

{| class="wikitable"
|-  align=center
|BGCOLOR="#ffc0c0"|111111
|BGCOLOR="#ffc0c0"|121212
|BGCOLOR="#ffc0c0"|111222
|BGCOLOR="#ffc0c0"|112122
|BGCOLOR="#c0c0ff"|111112(*)
|- align=center
|[[File:Uniform triangular tiling 111111.png|75px]]
|[[File:Uniform triangular tiling 121212.png|75px]]
|[[File:Uniform triangular tiling 111222.png|75px]]
|[[File:Uniform triangular tiling 112122.png|75px]]
|[[File:2-uniform_triangular_tiling_111112.png|75px]]
|- align=center
|p6m (*632)
|p3m1 (*333)
|cmm (2*22)
|p2 (2222)
|p2 (2222)
|}

{| class="wikitable"
|- align=center
|BGCOLOR="#ffc0c0"|121213
|BGCOLOR="#c0ffc0"|111212
|BGCOLOR="#c0ffc0"|111112
|BGCOLOR="#ffc0c0"|121314
|BGCOLOR="#c0ffc0"|111213
|-
|[[File:Uniform_triangular_tiling_121213.png|75px]]
|[[File:Uniform triangular tiling 111212.png|75px]]
|[[File:Uniform triangular tiling 111112.png|75px]]
|[[File:Uniform_triangular_tiling_121314.png|75px]]
|[[File:Uniform triangular tiling 111213.png|75px]]
|- align=center
|colspan=3|p31m (3*3)
|colspan=2|p3 (333)
|}

== A2 lattice and circle packings ==
{{distinguish|Strukturbericht designation#A-compounds{{!}}the A2 crystal lattice structure in the Strukturbericht classification system}}
[[File:Compound 3 triangular tilings.svg|thumb|The A{{sup sub|*|2}} lattice as three triangular tilings: {{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}}]]
The [[vertex arrangement]] of the triangular tiling is called an [[Root system#Explicit construction of the irreducible root systems|A<sub>2</sub> lattice]].<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html|title = The Lattice A2}}</ref> It is the 2-dimensional case of a [[simplectic honeycomb]].

The A{{sup sub|*|2}} lattice (also called A{{sup sub|3|2}}) can be constructed by the union of all three A<sub>2</sub> lattices, and equivalent to the A<sub>2</sub> lattice.
:{{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}} = dual of {{CDD|node_1|split1|branch_11}} = {{CDD|node_1|split1|branch}}

The vertices of the triangular tiling are the centers of the densest possible [[circle packing]].<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1</ref> Every circle is in contact with 6 other circles in the packing ([[kissing number]]). The packing density is {{frac|{{pi}}|{{sqrt|12}}}} or 90.69%. 
The [[voronoi cell]] of a triangular tiling is a [[hexagon]], and so the [[voronoi tessellation]], the hexagonal tiling, has a direct correspondence to the circle packings.
:[[File:1-uniform-11-circlepack.svg|200px]]

== Geometric variations  ==

Triangular tilings can be made with the equivalent {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces ([[Face-transitive|face-transitivity]]) and [[vertex-transitive|vertex-transitivity]], there are 5 variations. Symmetry given assumes all faces are the same color.<ref>''[[Tilings and Patterns]]'', from list of 107 isohedral tilings, p.473-481</ref>

<gallery>
Isohedral_tiling_p3-11.svg|[[Scalene triangle]]<BR/>p2 symmetry
Isohedral_tiling_p3-12.svg|Scalene triangle<BR/>pmg symmetry
Isohedral_tiling_p3-13.svg|[[Isosceles triangle]]<BR/>cmm symmetry
Isohedral_tiling_p3-11b.png|[[Right triangle]]<BR/>cmm symmetry
Isohedral_tiling_p3-14.svg|[[Equilateral triangle]]<BR/>p6m symmetry
</gallery>

== Related polyhedra and tilings ==

The planar tilings are related to [[polyhedra]]. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a [[pyramid (geometry)|pyramid]]. These can be expanded to [[Platonic solid]]s: five, four and three triangles on a vertex define an [[icosahedron]], [[octahedron]], and [[tetrahedron]] respectively.

This tiling is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,n}, continuing into the [[Hyperbolic space|hyperbolic plane]].
{{Triangular regular tiling}}

It is also topologically related as a part of sequence of [[Catalan solid]]s with [[face configuration]] Vn.6.6, and also continuing into the hyperbolic plane.
{| class="wikitable"
|-  align=center
|[[File:Triakistetrahedron.jpg|60px]]<BR/>[[Triakis tetrahedron|V3.6.6]]
|[[File:Tetrakishexahedron.jpg|60px]]<BR/>[[Tetrakis hexahedron|V4.6.6]]
|[[File:Pentakisdodecahedron.jpg|60px]]<BR/>[[Pentakis dodecahedron|V5.6.6]]
|[[File:Uniform polyhedron-63-t2.svg|60px]]<BR/>V6.6.6
|[[File:Heptakis heptagonal tiling.svg|60px]]<BR/>[[Order-7 truncated triangular tiling|V7.6.6]]
|}

=== Wythoff constructions from hexagonal and triangular tilings ===

Like the [[Uniform polyhedron|uniform polyhedra]] there are eight [[uniform tiling]]s that can be based from the regular hexagonal tiling (or the dual triangular tiling).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The ''truncated triangular tiling'' is topologically identical to the hexagonal tiling.)

{{Hexagonal tiling small table}}

{{Triangular tiling table}}

== Related regular complex apeirogons ==

There are 4 [[regular complex apeirogon]]s, sharing the vertices of the triangular tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons ''p''{''q''}''r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.<ref>Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.</ref>

The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.
{| class=wikitable
|-
|[[File:Complex apeirogon 2-6-6.png|160px]]
|[[File:Complex apeirogon 3-4-6.png|160px]]
|[[File:Complex apeirogon 3-6-3.png|160px]]
|[[File:Complex apeirogon 6-3-6.png|160px]]
|-
!2{6}6 or {{CDD|node_1|6|6node}}
!3{4}6 or {{CDD|3node_1|4|6node}}
!3{6}3 or {{CDD|3node_1|6|3node}}
!6{3}6 or {{CDD|6node_1|3|6node}}
|}

=== Other triangular tilings===
There are also three [[Laves tiling]]s made of single type of triangles:
{| class=wikitable
|- align=center valign=bottom
|[[File:1-uniform 3 dual.svg|240px]]<br/>[[Kisrhombille tiling|Kisrhombille]]<BR/>30°-60°-90° right triangles
|[[File:1-uniform 2 dual.svg|240px]]<br/>[[Tetrakis square tiling|Kisquadrille]]<BR/>45°-45°-90° right triangles
|[[File:1-uniform 4 dual.svg|240px]]<br/>[[Triakis triangular tiling|Kisdeltile]]<BR/>30°-30°-120° isosceles triangles
|}

==See also==
{{Commons category|Order-6 triangular tiling}}
* [[Triangular tiling honeycomb]]
* [[Simplectic honeycomb]]
* [[Tilings of regular polygons]]
* [[List of uniform tilings]]
* [[Isogrid]] (structural design using triangular tiling)

== References ==
{{Reflist}}

== Sources ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs
* {{cite book | author=Grünbaum, Branko | author-link=Branko Grünbaum | author2= Shephard, G. C. | name-list-style= amp | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: ''Regular and uniform tilings'', p.&nbsp;58-65, Chapter 2.9 Archimedean and Uniform colorings pp.&nbsp;102–107)
* {{The Geometrical Foundation of Natural Structure (book)}} p35
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]

== External links ==
* {{MathWorld | urlname=TriangularGrid | title=Triangular Grid}}
** {{MathWorld | urlname=RegularTessellation | title=Regular tessellation}}
** {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|x3o6o - trat - O2}}

{{Honeycombs}}
{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Isogonal tilings]]
[[Category:Isohedral tilings]]
[[Category:Regular tilings]]
[[Category:Triangular tilings| ]]
[[Category:Regular tessellations]]