# Triangle

Triangle | |
---|---|

Rank | 2 |

Type | Regular |

Notation | |

Bowers style acronym | Trig |

Coxeter diagram | x3o () |

Schläfli symbol | {3} |

Tapertopic notation | 1^{1} |

Elements | |

Edges | 3 |

Vertices | 3 |

Vertex figure | Dyad, length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 60° |

Height | |

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 | A_{2}, order 6 |

Flag orbits | 1 |

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 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 polyhedra, the great icosahedron. It is also used in many other polyhedra, including every one of the 92 Johnson solids.

## Naming[edit | edit source]

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".

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

## Vertex coordinates[edit | edit source]

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

- ,
- .

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

- .

## Representations[edit | edit source]

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[edit | edit source]

The equilateral triangle is seen in the vertex figures of four uniform polyhedra, including three Platonic solids and one Kepler-Poinsot polyhedron. It also appears in one Euclidean tiling.

Name | Picture | Edge lengths |
---|---|---|

Tetrahedron | 1 | |

Cube | √2 | |

Dodecahedron | (1+√5)/2 | |

Great stellated dodecahedron | (√5–1)/2 | |

Hexagonal tiling | √3 |

Besides these, equilateral triangles also appear as vertex figures in an infinite series of regular hyperbolic tilings.

## Other kinds of triangles[edit | edit source]

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[edit | edit source]

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

## External links[edit | edit source]

- 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".