Triangle

Triangle
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Trig
Coxeter diagram	x3o ()
Schläfli symbol	{3}
Tapertopic notation	11
Elements
Edges	3
Vertices	3
Vertex figure	Dyad, length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{3} ≈ 0.57735$
Inradius	$\frac{\sqrt3}{6} ≈ 0.28868$
Area	$\frac{\sqrt3}{4} ≈ 0.43301$
Angle	60°
Height	$\frac{\sqrt3}{2} ≈ 0.86603$
Central density	1
Number of external pieces	3
Level of complexity	1
Related polytopes
Army	Trig
Dual	Triangle
Conjugate	None
Abstract & topological properties
Flag count	6
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2, order 6
Convex	Yes
Net count	1
Nature	Tame

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.

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 an equilateral triangle of edge length 1 centered at the origin are:

• $\left(±\frac12,\,-\frac{\sqrt3}{6}\right),$
• $\left(0,\,\frac{\sqrt3}{3}\right).$

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

• $\left(\frac{\sqrt2}{2},\,0,\,0\right)$.

Representations

The triangle can be represented in three ways:

• x3o (full symmetry)
• ox&#x (axial, generally an isosceles triangle)
• ooo&#x (no symmetry, generally a scalene triangle)

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		1
Cube		√2
Dodecahedron		(1+√5)/2
Great stellated dodecahedron		(√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, has an inscribed and an circumscribed circle, and tiles the plane. These three properties don't generalize to any other n sided polygons.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Triangle".
• Hi.gher.Space Wiki Contributors. "Triangle".

• Hartley, Michael. "{3}*6".