geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the triangular tiling or triangular tessellation is one of the three

regular tilings This article lists the regular polytopes in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclide ...

of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

of English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter

delta Delta commonly refers to: * Delta (letter) (Δ or δ), the fourth letter of the Greek alphabet * D (NATO phonetic alphabet: "Delta"), the fourth letter in the Latin alphabet * River delta, at a river mouth * Delta Air Lines, a major US carrier ...

(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 In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

and the

hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a Truncation (geometry), truncated triangular tiling ...

Uniform colorings

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. 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.

A2 lattice and circle packings

The vertex arrangement of the triangular tiling is called an A₂ lattice. It is the 2-dimensional case of a simplectic honeycomb. The A lattice (also called A) can be constructed by the union of all three A₂ lattices, and equivalent to the A₂ lattice. : + + = dual of = The vertices of the triangular tiling are the centers of the densest possible

circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...

.Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1 Every circle is in contact with 6 other circles in the packing ( kissing number). The packing density is or 90.69%. The voronoi cell of a triangular tiling is a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

, and so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings. : 1-uniform-11-circlepack

Geometric variations

Triangular tilings can be made with the equivalent 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. Isohedral_tiling_p3-11.svg,

Scalene triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

p2 symmetry Isohedral_tiling_p3-12.svg, Scalene triangle
pmg symmetry Isohedral_tiling_p3-13.svg,

Isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

cmm symmetry Isohedral_tiling_p3-11b.png,

Right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

cmm symmetry Isohedral_tiling_p3-14.svg,

Equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

p6m symmetry

Related polyhedra and tilings

The planar tilings are related to

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a

pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

. These can be expanded to

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s: five, four and three triangles on a vertex define an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

, and

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

respectively. This tiling is topologically related as a part of sequence of regular polyhedra with

s , continuing into the hyperbolic plane. It is also topologically related as a part of sequence of

Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...

s with face configuration Vn.6.6, and also continuing into the hyperbolic plane.

Wythoff constructions from hexagonal and triangular tilings

Like the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...

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.)

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'r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136. The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.

Other triangular tilings

There are also three

Laves tiling This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean geometry, Euclidean plane, and their dual tilings. There are three regular and eight semiregular Tiling by regular polygons, tilings in the plane. The semi ...

s made of single type of triangles:

References

Sources

* Coxeter, H.S.M. '' Regular Polytopes'', (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs * (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65, Chapter 2.9 Archimedean and Uniform colorings pp. 102–107) * p35 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008,

External links

* ** ** * {{Tessellation Euclidean tilings Isogonal tilings Isohedral tilings Regular tilings Regular tessellations