Triangle

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Triangle
Regular triangle.svg
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymTrig
Coxeter diagramx3o (CDel node 1.pngCDel 3.pngCDel node.png)
Schläfli symbol{3}
Tapertopic notation11
Elements
Edges3
Vertices3
Vertex figureDyad, length 1
Measures (edge length 1)
Circumradius
Inradius
Area
Angle60°
Height
Central density1
Number of external pieces3
Level of complexity1
Related polytopes
ArmyTrig
DualTriangle
ConjugateNone
Abstract & topological properties
Flag count6
Euler characteristic0
OrientableYes
Properties
SymmetryA2, order 6
ConvexYes
Net count1
NatureTame

The triangle, also sometimes referred to as a trigon, is the simplest possible polygon, excluding the degenerate digon. Its highest symmetry version is called an equilateral triangle, to emphasize its three equal side lengths. It's the two dimensional simplex.

The combining prefix in BSAs is tr-, as in triddip.

The equilateral triangle is one of the only three regular polygons that can tile the plane, the other two being the square and the hexagon. Its tiling is called the triangular tiling, and it has 6 triangles per vertex, due to the angles of the triangle being 60°. It's also the regular simplex of highest dimension that can tile its respective (Euclidean) space.[1]

This is one of two polygons without a stellation, the other being the square, and one of three without a non-compound stellation, the third being the hexagon.

It is one of two possible segmentogons, being a point atop a dyad (that is, a dyadic pyramid). The other is the square.

The equilateral triangle is used as faces of 3 of the 5 Platonic solids, namely the tetrahedron, octahedron, and icosahedron, along with one of the Kepler-Poinsot solids, the great icosahedron. It is also used in many other polyhedra, including every one of the 92 Johnson solids.

Naming

The name triangle comes from Latin tres (3) and latin angulum (angle), referring to the number of sides. Alternate names include:

  • Trigon, from Ancient Greek τρεῖς (3) and γωνία (angle). More consistent with other polygons.
  • Trig, Bowers style acronym, short for "trigon".

Vertex coordinates

The vertices of a triangle of edge length 1 centered at the origin are:

Simpler coordinates can be given in three dimensions, as all permutations of:

  • .

Representations

The triangle can be represented in three ways:

In vertex figures

The equilateral triangle is seen in the vertex figures of four uniform polyhedra, including three Platonic Solids and one Kepler–Poinsot solid.

Equilateral triangles in vertex figures
Name Picture Edge lengths
Tetrahedron
Tetrahedron.png
1
Cube
Hexahedron.png
2
Dodecahedron
Dodecahedron.png
(1+5)/2
Great stellated dodecahedron
Great stellated dodecahedron.png
(5–1)/2

Other kinds of triangles

Beside the equilateral triangle, there are other kinds of triangles with non-equal edge lengths. These are the isosceles triangle with only two equal edge lengths, and the scalene triangle, with no equal edge lengths. Notably, these retain many of the properties of the highest-symmetry variant: any triangle is convex,[2] has an inscribed and an circumscribed circle,[3] and tiles the plane.[4] These three properties don't generalize to any other n sided polygons.

References

  1. Seng, Angina (January 25, 2019). "Regular tilings of n-simplex". Mathematics Stack Exchange.
  2. Matematleta (October 26, 2017). "Showing the inside of a triangle is a convex set".
  3. Weisstein, Eric W. "Triangle". MathWorld.
  4. Pegg Jr., Ed (March 7, 2011). "Any Triangle Can Tile". Wolfram Demonstrations Project.

External links