Triangular tiling

Triangular tiling
Type	Regular tiling
Vertex configuration	3.3.3.3.3.3 (or 36)
Face configuration	V6.6.6 (or V63)
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)
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

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 diagram		=
Schläfli symbol	{3,6}	h{6,3}	s{3,6}	s{3[3]}

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]]+

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 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.

*n32 symmetry mutation of regular tilings: {3,n}
Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
3.3	33	34	35	36	37	38	3∞	312i	39i	36i	33i

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)	[6,3+] (3*3)
{6,3}	t{6,3}	r{6,3}	t{3,6}	{3,6}	rr{6,3}	tr{6,3}	sr{6,3}	s{3,6}
63	3.122	(3.6)2	6.6.6	36	3.4.6.4	4.6.12	3.3.3.3.6	3.3.3.3.3.3
Uniform duals
V63	V3.122	V(3.6)2	V63	V36	V3.4.6.4	V.4.6.12	V34.6	V36

Wallpaper group	Triangle symmetry	Extended symmetry	Extended diagram	Extended group	Honeycombs
p3m1 (*333)	a1	[3[3]]		${\tilde {A}}_{2}$	(none)
p6m (*632)	i2	[[3[3]]] ↔ [6,3]	↔	${\tilde {A}}_{2}\times 2\leftrightarrow {\tilde {G}}_{2}$	1, 2
p31m (3*3)	g3	[3+[3[3]]] ↔ [6,3+]	↔	${\tilde {A}}_{2}\times 3\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}$	(none)
p6 (632)	r6	[3[3[3]]]+ ↔ [6,3]+	↔	${\tfrac {1}{2}}{\tilde {A}}_{2}\times 6\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}$	(1)
p6m (*632)		[3[3[3]]] ↔ [6,3]		${\tilde {A}}_{2}\times 6\leftrightarrow {\tilde {G}}_{2}$	3

Triangular symmetry tilings
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.

• 
  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

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 0-7167-1193-1.{{cite book}}: CS1 maint: multiple names: authors list (link) (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

• Weisstein, Eric W. "Triangular Grid". MathWorld.
  • Weisstein, Eric W. "Regular tessellation". MathWorld.
  • Weisstein, Eric W. "Uniform tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings x3o6o - trat - O2".

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21