# Tetrahedron

In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length.

The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges:

Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are:

The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron

The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion.

This property also applies for tetragonal disphenoids when applied to the two special edge pairs.

Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.

Kepler's drawing of a regular tetrahedron inscribed in a cube, and one of the four trirectangular tetrahedra that surround it, filling the cube.
For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.

The absolute value of the scalar triple product can be represented as the following absolute values of determinants:

Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant:

The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.

An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.

A tetrahedron is a 3-simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space).

The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, five is the minimum number of tetrahedra required to compose a cube. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be added to make a cube, which has 8 vertices.

Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra.

If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.)

The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.

A law of sines for tetrahedra and the space of all shapes of tetrahedra

One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.

Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.

In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron. Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron.

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures.

The square hosohedron is another polyhedron with four faces, but it does not have triangular faces.

The Szilassi polyhedron and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face.

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes.

Quaternary phase diagrams of mixtures of chemical substances are represented graphically as tetrahedra.

However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.

The Royal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice.

Modern caltrop with hollow spikes to puncture self-sealing rubber tyres

Some caltrops are based on tetrahedra as one spike points upwards regardless of how they land and can be easily made by welding two bent nails together.