# Triangular tiling

Triangular tiling

Type Regular tiling
Vertex configuration 3.3.3.3.3.3 (or 36)
Schläfli symbol(s) {3,6}
{3[3]}
Wythoff symbol(s) 6 | 3 2
3 | 3 3
| 3 3 3
Coxeter diagram(s)

=
Symmetry p6m, [6,3], (*632)
p3m1, [3[3]], (*333)
p3, [3[3]]+, (333)
Rotation symmetry p6, [6,3]+, (632)
p3, [3[3]]+, (333)
Dual Hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings 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 calls 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

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)

Four of the colorings are generated by Wythoff constructions. 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 Coloring Symmetry Wythoff symbol indices 111111 121212 121213 121314 *632 p6m [6,3] *333 p3m1 [3[3]] = [1+,6,3] 3*3 p31m [6,3+] 333 p3 [3[3]]+ 6 | 3 2 3 | 3 3 | 3 3 3 = {3,6} h{6,3} s{3,6} s{3[3]}
 Coloring Coloring Symmetry indices 111222 112122 111112 111212 111213 2*22 cmm [∞,2+,∞] 2222 p2 [∞,2,∞]+ *333 p3m1 [3[3]] *333 p3m1 [3[3]] 333 p3 [3[3]]+

## A2 lattice and circle packings

The A*
2
lattice as three triangular tilings: + +

The vertex arrangement of the triangular tiling is called an A2 lattice.[1] It is the 2-dimensional case of a simplectic honeycomb.

The A*
2
lattice (also called A3
2
) 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. 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%. Since the union of 3 A2 lattices is also an A2 lattice, the circle packing can be given with 3 colors of circles.

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.

A2 lattice circle packing A*
2
lattice circle packing
Hexagonal tilings

## 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.

Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
{3,2} {3,3} {3,4} {3,5} {3,6} {3,7} {3,8} (3,∞} {3,12i} {3,9i} {3,6i} {3,3i}

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.

 V3.6.6 V4.6.6 V5.6.6 V6.6.6 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.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[1+,6,3]
(*333)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}

=

=

=
=
or
=
or

=

Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36

Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[3]] ×1 (None)
i2 <[3[3]]>
= [6,3]

=
×2 1, 2
r6 [3[3[3]]]
= [6,3]

=
×6 3, (1)
Wythoff 3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 3 3 | | 3 3 3
Coxeter
Image
Vertex figure

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

6.6.6

3.3.3.3.3.3

## Triangular tiling variations

Triangular tilings can be made with the identical {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.[2]