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Template:Uniform tiles db In geometry, the triangular tiling 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 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.

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 indices	111111	121212	121213	121314
Coloring
Symmetry	*632 (p6m) [6,3]	*333 (p3m1) [3[3]] = [1+,6,3]	3*3 (p31m) [6,3+]	333 (p3) [3[3]]+
Wythoff symbol	6 | 3 2	3 | 3 3		| 3 3 3
Coxeter-Dynkin	Template:CDD	Template:CDD = Template:CDD	Template:CDD	Template:CDD

Coloring indices	111222	112122	111112	111212	111213
Coloring	File:Uniform triangular tiling 111222.png	File:Uniform triangular tiling 112122.png	File:Uniform triangular tiling 111112.png	File:Uniform triangular tiling 111212.png	File:Uniform triangular tiling 111213.png
Symmetry	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 vertex arrangement of the triangular tiling is called an A₂ lattice. It is the 2-dimensional case of a simplectic honeycomb.

The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) can be constructed by the union of all three A₂ lattices, and equivalent to the A₂ lattice.

Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD = Template:CDD

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	ATemplate:Sup sub lattice circle packing
File:Triangular tiling circle packing.png	File:Triangular tiling circle packing3.png
Hexagonal tilings
File:Uniform tiling 63-t0.png	File:Uniform tiling 333-t012.png

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. Template: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.

File:Triakistetrahedron.jpg V3.6.6

File:Tetrakishexahedron.jpg V4.6.6

File:Pentakisdodecahedron.jpg V5.6.6

File:Uniform polyhedron-63-t2.png V6.6.6

File:Order3 heptakis heptagonal til.png 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.)

46 year old Fitter (Normal ) Anton from Roberval, has numerous pursuits including handwriting analysis, property developers in singapore and handball. These days had buying A property in singapore family journey to Pearling. Template:Triangular tiling table

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.

• 
  Scalene triangle
  p2 symmetry
• 
  Scalene triangle
  pmg symmetry
• 
  Isosceles triangle
  cmm symmetry
• 
  Right triangle
  cmm symmetry
• 
  Equilateral triangle
  p6m symmetry

See also

• Triangular tiling honeycomb
• Simplectic honeycomb
• Tilings of regular polygons
• List of uniform tilings

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• Template:The Geometrical Foundation of Natural Structure (book) p35
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]

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