Triangular tiling

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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) CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node 1.pngCDel split1.pngCDel branch.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel split1.pngCDel branch hh.png
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 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[edit]

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 Uniform tiling 63-t2.png Uniform tiling 333-t1.png Uniform tiling 63-h12.png Uniform tiling 333-snub.png
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 diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png CDel node 1.pngCDel split1.pngCDel branch.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.png CDel node h.pngCDel split1.pngCDel branch hh.png
Schläfli symbol {3,6} h{6,3} s{3,6} s{3[3]}
Coloring
indices
111222 112122 111112 111212 111213
Coloring Uniform triangular tiling 111222.png Uniform triangular tiling 112122.png Uniform triangular tiling 111112.png Uniform triangular tiling 111212.png 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[edit]

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.

CDel node 1.pngCDel split1.pngCDel branch.png + CDel node.pngCDel split1.pngCDel branch 10lu.png + CDel node.pngCDel split1.pngCDel branch 01ld.png = dual of CDel node 1.pngCDel split1.pngCDel branch 11.png = CDel node 1.pngCDel split1.pngCDel branch.png

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
Triangular tiling circle packing.png Triangular tiling circle packing3.png
Hexagonal tilings
Uniform tiling 63-t0.png Uniform tiling 333-t012.png

Related polyhedra and tilings[edit]

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.

Finite Euclidean Compact hyperbolic Paracompact
Trigonal dihedron.png
{3,2}
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-43-t2.png
{3,4}
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-63-t2.png
{3,6}
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 39-t0.png
{3,9}
... H2 tiling 23i-4.png
(3,∞}

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.

Triakistetrahedron.jpg
V3.6.6
Tetrakishexahedron.jpg
V4.6.6
Pentakisdodecahedron.jpg
V5.6.6
Uniform polyhedron-63-t2.png
V6.6.6
Order3 heptakis heptagonal til.png
V7.6.6

Wythoff constructions from hexagonal and triangular tilings[edit]

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]}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png =
CDel branch 10ru.pngCDel split2.pngCDel node.png or CDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png
Uniform tiling 333-t01.png
Uniform tiling 63-t12.png
Uniform tiling 333-t012.png
Uniform tiling 63-t2.png
Uniform tiling 333-t2.png
Uniform tiling 63-t02.png
Rhombitrihexagonal tiling snub edge coloring.png
Uniform tiling 63-t012.png Uniform tiling 63-snub.png Uniform tiling 333-t0.pngUniform tiling 333-t1.png Uniform tiling 333-t02.pngUniform tiling 333-t12.png Uniform tiling 63-h12.png
Uniform tiling 333-snub.png
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
CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t2.png Uniform tiling 63-t0.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Uniform tiling 63-t0.png Rhombic star tiling.png Uniform tiling 63-t0.png


Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[3]] CDel node.pngCDel split1.pngCDel branch.png ×1 (None)
i2 <[3[3]]>
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c2.png
= CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node.png
×2 CDel node 1.pngCDel split1.pngCDel branch.png 1, CDel node.pngCDel split1.pngCDel branch 11.png 2
r6 [3[3[3]]]
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c1.png
= CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
×6 CDel node 1.pngCDel split1.pngCDel branch 11.png 3, CDel node h.pngCDel split1.pngCDel branch hh.png (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 CDel node 1.pngCDel split1.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 11.png CDel node.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 11.png CDel node h.pngCDel split1.pngCDel branch hh.png
Image
Vertex figure
Uniform tiling 333-t0.png
(3.3)3
Uniform tiling 333-t01.png
3.6.3.6
Uniform tiling 333-t1.png
(3.3)3
Uniform tiling 333-t12.png
3.6.3.6
Uniform tiling 333-t2.png
(3.3)3
Uniform tiling 333-t02.png
3.6.3.6
Uniform tiling 333-t012.png
6.6.6
Uniform tiling 333-snub.png
3.3.3.3.3.3

Triangular tiling variations[edit]

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]

See also[edit]

Notes[edit]

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html
  2. ^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481

References[edit]

External links[edit]