Triangular tiling | |
---|---|

| |

Type | Regular tiling |

Vertex configuration | 3.3.3.3.3.3 (or 3^{6}) |

Face configuration | V6.6.6 (or V6^{3}) |

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** or **triangular tessellation** is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. 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}.

- Uniform colorings
- A2 lattice and circle packings
- Geometric variations
- Related polyhedra and tilings
- Wythoff constructions from hexagonal and triangular tilings
- Related regular complex apeirogons
- Other triangular tilings
- See also
- References
- Sources
- External links

English mathematician John Conway called 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.

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) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.^{ [1] }

There is one class of Archimedean colorings, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.

111111 | 121212 | 111222 | 112122 | 111112(*) |

p6m (*632) | p3m1 (*333) | cmm (2*22) | p2 (2222) | p2 (2222) |

121213 | 111212 | 111112 | 121314 | 111213 |

p31m (3*3) | p3 (333) |

The vertex arrangement of the triangular tiling is called an A_{2} lattice.^{ [2] } It is the 2-dimensional case of a simplectic honeycomb.

The A^{*}_{2} lattice (also called A^{3}_{2}) can be constructed by the union of all three A_{2} lattices, and equivalent to the A_{2} lattice.

- + + = dual of =

The vertices of the triangular tiling are the centers of the densest possible circle packing.^{ [3] } Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is π⁄√12 or 90.69%. 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.

Triangular tilings can be made with the equivalent {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.^{ [4] }

- Scalene triangle

p2 symmetry - Scalene triangle

pmg symmetry - Isosceles triangle

cmm symmetry - Right triangle

cmm symmetry - Equilateral triangle

p6m symmetry

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 | 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 |

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 | ||||||||
---|---|---|---|---|---|---|---|---|

Fundamental domains | Symmetry: [6,3], (*632) | [6,3]^{+}, (632) | ||||||

{6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} | |

Config. | 6^{3} | 3.12.12 | (6.3)^{2} | 6.6.6 | 3^{6} | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |

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 |

There are 4 regular complex apeirogons, sharing the vertices of the triangular tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons *p*{*q*}*r* are constrained by: 1/*p* + 2/*q* + 1/*r* = 1. Edges have *p* vertices, and vertex figures are *r*-gonal.^{ [5] }

The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.

2{6}6 or | 3{4}6 or | 3{6}3 or | 6{3}6 or |
---|

There are also three Laves tilings made of single type of triangles:

Kisrhombille 30°-60°-90° right triangles | Kisquadrille 45°-45°-90° right triangles | Kisdeltile 30°-30°-120° isosceles triangles |

Wikimedia Commons has media related to Order-6 triangular tiling .

- Triangular tiling honeycomb
- Simplectic honeycomb
- Tilings of regular polygons
- List of uniform tilings
- Isogrid (structural design using triangular tiling)

In geometry, the **truncated octahedron** is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a **6**-zonohedron. It is also the Goldberg polyhedron G_{IV}(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

In geometry, the **hexagonal tiling** or **hexagonal tessellation** is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or *t*{3,6} .

In geometry, the **square tiling**, **square tessellation** or **square grid** is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a **quadrille**.

In geometry, the **truncated hexagonal tiling** is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.

In geometry, the **truncated trihexagonal tiling** is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of *tr*{3,6}.

In geometry, the **rhombitrihexagonal tiling** is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

In geometry, the **trihexagonal tiling** is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.

In geometry, the **snub hexagonal tiling** is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol *sr{3,6}*. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol *sr{4,6}*.

In geometry, the **elongated triangular tiling** is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a **cubille**.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The **bitruncated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

In geometry, the **truncated heptagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of *t*{7,3}. The tiling has a vertex configuration of 3.14.14.

In geometry, the **rhombitriheptagonal tiling** is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

In geometry, the **order-7 truncated triangular tiling**, sometimes called the **hyperbolic soccerball**, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.

In geometry, the **octagonal tiling** is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of *{8,3}*, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

In geometry, the **rhombitrioctagonal tiling** is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

In geometry, the **order-3 apeirogonal tiling** is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

In geometry, the **order-6 apeirogonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

- ↑
*Tilings and patterns*, p.102-107 - ↑ "The Lattice A2".
- ↑ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1
- ↑
*Tilings and Patterns*, from list of 107 isohedral tilings, p.473-481 - ↑ Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.

- Coxeter, H.S.M.
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs - Grünbaum, Branko & Shephard, G. C. (1987).
*Tilings and Patterns*. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1:*Regular and uniform tilings*, p. 58-65, Chapter 2.9 Archimedean and Uniform colorings pp. 102–107) - 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-Strauss,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5

- Weisstein, Eric W. "Triangular Grid".
*MathWorld*. - Klitzing, Richard. "2D Euclidean tilings x3o6o - trat - O2".

Space | Family | / / | ||||
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E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |

E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |

E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |

E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |

E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |

E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |

E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |

E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |

E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |

E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |

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