In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the triangular tiling is one of the three regular tiling

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

s of the Euclidean plane. 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 {3,6}.

Conway

John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

 calls it a deltille, named from the triangular shape of the greek letter delta (Δ). The triangular tiling is roughly the kishextile.

It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.

This vertex arrangement

Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....

 is called the A2 lattice.

Uniform colorings

There are 9 distinct uniform coloring

Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive...

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)

Four of the colorings are generated by Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

s. Seven of the nine distinct colorings can be made as reductions of the four coloring: 121314. The remaining two, 111222 and 112122, have no Wythoff constructions.

Coloring indices	111111	121212	121314	121213
Coloring
Symmetry	*632 (p6m) [6,3]	*333 (p3m1) [3[3]]	333 (p3) [3[3]]+	3*3 (p31m) [6,3+]
Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....	6 | 3 2	3 | 3 3	| 3 3 3
Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Coloring indices	111222	112122	111112	111212	111213
Coloring
Symmetry	2*22 (cmm) [∞,2+,∞]	2222 (p2) [∞,2,∞]+	*333 (p3m1) [3[3]]	*333 (p3m1) [3[3]]	333 (p3) [3[3]]+

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

Pyramid

A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...

. These can be expanded to Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

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

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

, and tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 respectively.

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane

Hyperbolic space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

.

{3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

{3,4} Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

{3,5} Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

{3,6}

{3,7}

{3,8}

{3,9}

It is also topologically related as a part of sequence of Catalan solid

Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

s with face configuration

Face configuration

In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....

 Vn.6.6, and also continuing into the hyperbolic plane.

V3.6.6 Triakis tetrahedron In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron...

V4.6.6 Tetrakis hexahedron In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....

V5.6.6 Pentakis dodecahedron In geometry, a pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron...

V6.6.6

V7.6.6 Order-7 truncated triangular tiling In geometry, the Order 7 truncated heptagonal tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball with heptagons in place of pentagons...

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