Triangular tiling


Triangular tiling
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-Dynkin(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 h.pngCDel split1.pngCDel branch hh.png
Symmetry p6m, [6,3], *632
p3m1, [3[3]], *333
p3, [3[3]]+, 333
Dual Hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive
Triangular tiling
3.3.3.3.3.3 (or 36)

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.

This vertex arrangement is called the A2 lattice.[1]

Contents

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 121314 121213
Coloring Uniform tiling 63-t2.png Uniform tiling 333-t1.png Uniform tiling 333-snub.png Uniform tiling 63-h12.png
Symmetry *632
(p6m)
[6,3]
*333
(p3m1)
[3[3]]
333
(p3)
[3[3]]+
3*3
(p31m)
[6,3+]
Wythoff symbol 6 | 3 2 3 | 3 3 | 3 3 3
Coxeter-Dynkin CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel split1.pngCDel branch.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel split1.pngCDel branch hh.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
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]]+

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.

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}

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

See also

Notes

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
  • Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1.  (Chapter 2.1: Regular and uniform tilings, p.58-65)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p35
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]

External links


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