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| {{Uniform tiles db|Reg tiling stat table|Ut}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. The preferred pastime for him and his kids is to perform lacross and he'll be beginning something else along with it. I've usually cherished residing in Alaska. Distributing production is how he makes a residing.<br><br>Also visit my blog: [http://help.ksu.edu.sa/node/65129 online reader] |
| In [[geometry]], the '''triangular tiling''' is one of the three regular [[tessellation|tiling]]s 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}.
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| [[John Horton Conway|Conway]] calls it a '''deltille''', named from the triangular shape of the Greek letter delta (Δ). The triangular tiling is roughly the kishextile.
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| It is one of [[List_of_regular_polytopes#Euclidean_tilings|three regular tilings of the plane]]. The other two are the [[square tiling]] and the [[hexagonal tiling]].
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| == Uniform colorings ==
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| There are 9 distinct [[uniform coloring]]s 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)
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| Four of the colorings are generated by [[Wythoff construction]]s. 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.
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| {| class="wikitable"
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| |- align=center
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| !Coloring<BR>indices
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| |111111
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| |121212
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| |121213
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| |121314
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| |- align=center
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| !Coloring
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| |[[File:Uniform tiling 63-t2.png|60px]]
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| |[[File:Uniform tiling 333-t1.png|60px]]
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| |[[File:Uniform tiling 63-h12.png|60px]]
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| |[[File:Uniform tiling 333-snub.png|60px]]
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| |- align=center
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| ![[List of planar symmetry groups|Symmetry]]
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| |*632<BR>(p6m)<BR>[6,3]
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| |*333<BR>(p3m1)<BR>[3<sup>[3]</sup>] = [1<sup>+</sup>,6,3]
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| |3*3<BR>(p31m)<BR>[6,3<sup>+</sup>]
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| |333<BR>(p3)<BR>[3<sup>[3]</sup>]<sup>+</sup>
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| |- align=center
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| ![[Wythoff symbol]]
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| |6 | 3 2
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| |3 | 3 3
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| || 3 3 3
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| |- align=center
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| ![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
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| |{{CDD|node|6|node|3|node_1}}
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| |{{CDD|node_1|split1|branch}} = {{CDD|node_h1|6|node|3|node}}
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| |{{CDD|node|6|node_h|3|node_h}}
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| |{{CDD|node_h|split1|branch_hh}}
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| |}
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| {| class="wikitable"
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| !Coloring<BR>indices
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| |111222
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| |112122
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| |111112
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| |111212
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| |111213
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| |-
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| !Coloring
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| |[[File:Uniform triangular tiling 111222.png|60px]]
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| |[[File:Uniform triangular tiling 112122.png|60px]]
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| |[[File:Uniform triangular tiling 111112.png|60px]]
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| |[[File:Uniform triangular tiling 111212.png|60px]]
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| |[[File:Uniform triangular tiling 111213.png|60px]]
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| |- align=center
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| ![[List of planar symmetry groups|Symmetry]]
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| |2*22<BR>(cmm)<BR>[∞,2<sup>+</sup>,∞]
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| |2222<BR>(p2)<BR>[∞,2,∞]<sup>+</sup>
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| |*333<BR>(p3m1)<BR>[3<sup>[3]</sup>]
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| |*333<BR>(p3m1)<BR>[3<sup>[3]</sup>]
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| |333<BR>(p3)<BR>[3<sup>[3]</sup>]<sup>+</sup>
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| |}
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| == A2 lattice and circle packings ==
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| The [[vertex arrangement]] of the triangular tiling is called an [[A2 lattice|A<sub>2</sub> lattice]].<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html</ref> It is the 2-dimensional case of a [[simplectic honeycomb]].
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| The A{{sup sub|*|2}} lattice (also called A{{sup sub|3|2}}) can be constructed by the union of all three A<sub>2</sub> lattices, and equivalent to the A<sub>2</sub> lattice.
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| :{{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}} = dual of {{CDD|node_1|split1|branch_11}} = {{CDD|node_1|split1|branch}}
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| 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 <math>\frac{\pi}{\sqrt{12}}</math> or 90.69%. Since the union of 3 A<sub>2</sub> lattices is also an A<sub>2</sub> lattice, the circle packing can be given with 3 colors of circles.
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| 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.
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| {| class=wikitable
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| !A<sub>2</sub> lattice circle packing
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| !A{{sup sub|*|2}} lattice circle packing
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| |-
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| |[[File:triangular tiling circle packing.png|180px]]
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| |[[File:Triangular tiling circle packing3.png|180px]]
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| |-
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| !colspan=2|[[Hexagonal tiling]]s
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| |-
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| |[[File:Uniform tiling 63-t0.png|180px]]
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| |[[File:Uniform tiling 333-t012.png|180px]]
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| |}
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| == Related polyhedra and tilings ==
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| 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 solid]]s: five, four and three triangles on a vertex define an [[icosahedron]], [[octahedron]], and [[tetrahedron]] respectively.
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| This tiling is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,n}, continuing into the [[Hyperbolic space|hyperbolic plane]].
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| {{Triangular regular tiling}}
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| It is also topologically related as a part of sequence of [[Catalan solid]]s with [[face configuration]] Vn.6.6, and also continuing into the hyperbolic plane.
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| {| class="wikitable"
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| |- align=center
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| |[[File:Triakistetrahedron.jpg|60px]]<BR>[[Triakis tetrahedron|V3.6.6]]
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| |[[File:Tetrakishexahedron.jpg|60px]]<BR>[[Tetrakis hexahedron|V4.6.6]]
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| |[[File:Pentakisdodecahedron.jpg|60px]]<BR>[[Pentakis dodecahedron|V5.6.6]]
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| |[[File:Uniform polyhedron-63-t2.png|60px]]<BR>V6.6.6
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| |[[File:Order3 heptakis heptagonal til.png|60px]]<BR>[[Order-7 truncated triangular tiling|V7.6.6]]
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| |}
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| === Wythoff constructions from hexagonal and triangular tilings ===
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| Like the [[Uniform polyhedron|uniform polyhedra]] there are eight [[uniform tiling]]s that can be based from the regular hexagonal tiling (or the dual triangular tiling).
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| 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.)
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| {{Hexagonal tiling table}}
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| {{A2_honeycombs}}
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| {{Triangular tiling table}}
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| == Triangular tiling variations ==
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| Triangular tilings can be made with the identical {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces ([[Face-transitive|face-transitivity]]) and [[vertex-transitive|vertex-transitivity]], there are 5 variations. Symmetry given assumes all faces are the same color.<ref>Tilings and Patterns, from list of 107 isohedral tilings, p.473-481</ref>
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| <gallery>
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| Isohedral_tiling_p3-11.png|[[Scalene triangle]]<BR>p2 symmetry
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| Isohedral_tiling_p3-12.png|Scalene triangle<BR>pmg symmetry
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| Isohedral_tiling_p3-13.png|[[Isosceles triangle]]<BR>cmm symmetry
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| Isohedral_tiling_p3-11b.png|[[Right triangle]]<BR>cmm symmetry
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| Isohedral_tiling_p3-14.png|[[Equilateral triangle]]<BR>p6m symmetry
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| </gallery>
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| ==See also==
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| {{Commonscat|Order-6 triangular tiling}}
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| * [[Triangular tiling honeycomb]]
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| * [[Simplectic honeycomb]]
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| * [[Tilings of regular polygons]]
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| * [[List of uniform tilings]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
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| * {{cite book|author=[[Branko Grünbaum|Grünbaum, Branko]] ; and Shephard, G. C.| title=Tilings and Patterns| location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1}} (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65)
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| * {{The Geometrical Foundation of Natural Structure (book)}} p35
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| * John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 [http://www.akpeters.com/product.asp?ProdCode=2205]
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| == External links ==
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| * {{MathWorld | urlname=TriangularGrid | title=Triangular Grid}}
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| ** {{MathWorld | urlname=RegularTessellation | title=Regular tessellation}}
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| ** {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
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| * {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|x3o6o - trat - O2}}
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| {{Honeycombs}}
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| [[Category:Tessellation]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. The preferred pastime for him and his kids is to perform lacross and he'll be beginning something else along with it. I've usually cherished residing in Alaska. Distributing production is how he makes a residing.
Also visit my blog: online reader