{{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 regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. 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 Conway called it a '''deltille''', named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a '''kishextille''' by a kis operation that adds a center point and triangles to replace the faces of a hextille.

It is one of 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 colorings 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 colorings, 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 |75px |75px |75px |75px |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 |- |75px |75px |75px |75px |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}} 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 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 simplicial 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. :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-transitivity) and 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.svg|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. These can be expanded to Platonic solids: 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 symbols {3,n}, continuing into the hyperbolic plane. {{Triangular regular tiling}}

It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also continuing into the hyperbolic plane. {| class="wikitable" |- align=center |60px<BR/>V3.6.6 |60px<BR/>V4.6.6 |60px<BR/>V5.6.6 |60px<BR/>V6.6.6 |60px<BR/>V7.6.6 |}

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

Like the uniform polyhedra there are eight uniform tilings 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 apeirogons, 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 |- |160px |160px |160px |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 tilings made of single type of triangles: {| class=wikitable |- align=center valign=bottom |240px<br/>Kisrhombille<BR/>30°-60°-90° right triangles |240px<br/>Kisquadrille<BR/>45°-45°-90° right triangles |240px<br/>Kisdeltile<BR/>30°-30°-120° isosceles triangles |}

==See also== * Isogrid (structural design using triangular tiling) * List of uniform tilings * Simplectic honeycomb * Tilings of regular polygons * Triangular tiling honeycomb

== References == {{Reflist}}

== Sources == * Coxeter, H.S.M. ''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 == {{commonscatinline}} * {{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