Triangle (EntityTopic, 17)

From Hi.gher. Space

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{{Shape
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<[#ontology [kind topic] [cats 2D Simplex] [alt [[freebase:07jx7]] [[wikipedia:Triangle]]]]>
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| attrib=pure
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{{STS Shape
| dim=2
| dim=2
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| elements=3, 3
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| elements=3 [[digon]]s, 3 [[point]]s
| genus=0
| genus=0
| 20=SSC
| 20=SSC
| ssc=G3
| ssc=G3
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| rns=1<sup>1</sup> x<sup>y</sup>
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| ssc2=G3
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| rot_i=3
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| pv_circle=<sup>[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>π</sub> ≈ 0.2757
| pv_circle=<sup>[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>π</sub> ≈ 0.2757
| pv_square=[[Sine values|S<sub>1</sub>]]+S<sub>1</sub><sup>3</sup>S<sub>5</sub><sup>-2</sup> ≈ 0.2774
| pv_square=[[Sine values|S<sub>1</sub>]]+S<sub>1</sub><sup>3</sup>S<sub>5</sub><sup>-2</sup> ≈ 0.2774
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| extra={{STS Matrix|
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3 0
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1 1}}{{STS Tapertope
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| order=1, 1
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| notation=1<sup>1</sup>
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| index=4
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}}{{STS Polytope
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| bowers=Trig
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| dual=''Self-dual''}}{{STS Uniform polytope
| schlaefli={3}
| schlaefli={3}
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| vfigure=[[Line segment]], length 1
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| dynkin=x3o
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| vfigure=[[Digon]], length 1
| dual=''Self-dual''
| dual=''Self-dual''
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}}
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}}}}
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A '''triangle''' is a two-dimensional [[simplex]].
A '''triangle''' is a two-dimensional [[simplex]].
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The triangle is important as all [[polygon]]s can be defined as the [[union]] of a [[set]] of triangles, and all [[polytope]]s in greater than two dimensions can be defined as the [[hypervolume]] bounded by the union of a set of triangles.
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Triangles are one of the three regular polygons that can tile the plain, forming the [[triangular tiling]].
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== Equations ==
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==Coordinates==
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*The area of an '''arbitrary''' triangle is:
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The Cartesian coordinates of an equilateral triangle centered at the origin with side length 2 are:
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<blockquote>''ab''sin(''C'')2<sup>-1</sup></blockquote>
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*where ''a'' and ''b'' are two sides of the triangle and ''C'' is the angle between them.
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== Segmentation ==
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(-1, -√3/3), (1, -√3/3), (0, 2√3/3)
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The equilateral triangle of side 1 may be [[segment]]ed into 3× triangle with sides 1, 2×3<sup>-2<sup>-1</sup></sup> and angles 2×30°, 120°.
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== Equations ==
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*The area of an equilateral triangle with side length ''l'' is:
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<blockquote>{{Over|√3|4}} {{DotHV}}</blockquote>
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*The area of an arbitrary triangle, where ''a'' and ''b'' are two sides of the triangle and ''C'' is the angle between them, is:
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<blockquote><sup>1</sup>∕<sub>2</sub> &middot; ''ab'' &middot; sin ''C''</blockquote>
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== Use ==
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== Dissection ==
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Triangular faces are found in these trishapes on FGwiki:
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The equilateral triangle of side 1 may be [[dissect]]ed into 3× triangle with sides 1, √3/3, √3/3 and angles 30°, 30°, 120°.
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*[[Icosahedron]] (20×, 100%)
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*[[Octahedron]] (8×, 100%)
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*[[Tetrahedron]] (4×, 100%)
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*[[Dodecahedral snub]] (80×, 87%)
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*[[Cubic snub]] (32×, 84%)
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*[[Square pyramid]] (4×, 80%)
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*[[Icosidodecahedron]] (20×, 63%)
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*[[Dodecahedral truncate]] (20×, 63%)
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*[[Cuboctahedron]] (8×, 57%)
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*[[Cubic truncate]] (8×, 57%)
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*[[Tetrahedral truncate]] (4×, 50%)
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*[[Triangular prism]] (2×, 40%)
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Triangular segments are found in these dishapes on FGwiki:
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<[#polytope [id -3]]>
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*[[Hexagon]] (6×)
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{{Simplices|2}}
{{Dishapes}}
{{Dishapes}}
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{{Rotope Nav|2|3|4|II<br>Square|I'<br>Triangle|(II)<br>Circle|gons}}
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{{Tapertope Nav|3|4|5|11<br>Square|1<sup>1</sup><br>Triangle|3<br>Sphere|gons}}
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[[Category:Regular polygons]]
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Latest revision as of 11:40, 26 March 2017

A triangle is a two-dimensional simplex.

Triangles are one of the three regular polygons that can tile the plain, forming the triangular tiling.

Coordinates

The Cartesian coordinates of an equilateral triangle centered at the origin with side length 2 are:

(-1, -√3/3), (1, -√3/3), (0, 2√3/3)

Equations

  • The area of an equilateral triangle with side length l is:
√34 · l2
  • The area of an arbitrary triangle, where a and b are two sides of the triangle and C is the angle between them, is:
12 · ab · sin C

Dissection

The equilateral triangle of side 1 may be dissected into 3× triangle with sides 1, √3/3, √3/3 and angles 30°, 30°, 120°.

Incidence matrix

Dual: Self-dual

#TXIDVaEaTypeName
0 Va= point ;
1 Ea2= digon ;
2 3a33= triangle ;

Usage as facets


Simplices
triangletetrahedronpyrochoronpyroteronpyropeton


Notable Dishapes
Flat: trianglesquarepentagonhexagonoctagondecagon
Curved: circle


3. 11
Square
4. 11
Triangle
5. 3
Sphere
List of tapertopes

Pages in this category (1)