# Tetrahedron

Regular tetrahedron
Tetrahedron
Click on picture for large version.
Type Platonic
Face polygon triangle
Faces 4
Edges 6
Vertices 4
Faces per vertex 3
Vertices per face 3
Symmetry group tetrahedral (Td)
of order 24
Dual polyhedron tetrahedron (self-dual)
Dihedral angle 70° 32' = arccos(1/3)
Properties regular, convex

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids.

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

## Area and volume

The area A and the volume V of a regular tetrahedron of edge length a are:

$\displaystyle A=\sqrt{3}a^2$
$\displaystyle V=\begin{matrix}{1\over12}\end{matrix}\sqrt{2}a^3$

The height is $\displaystyle h=(a/3) \sqrt{6}$ , the angle between an edge and a face is arctan $\displaystyle \sqrt{2}$ (ca. 55°), and between two faces arccos (1/3) = arctan $\displaystyle 2\sqrt{2}$ (ca. 71°). Note that with respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that in a face, from the midpoint at the base.

Like for any pyramid, the volume is $\displaystyle V = \frac{1}{3} Ah$ where A is the area of the base and h the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apexes to the opposite faces are inversely proportional to the areas of these faces.

Also, for a tetrahedron ABCT the volume is given by

$\displaystyle V = \frac {AT \cdot BT \cdot CT}{6} \cdot \sqrt {1 + 2 \cdot \cos a \cdot \cos b \cdot \cos c - \cos^2 a - \cos^2 b - \cos^2 c}$

where a is angle ATB, b angle BTC, and c angle CTB.

For the distance between edges, see skew line.

The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(ab, bc, cd)|, or any other combination of pairs of vertices that form a simply connected graph.

## Geometric relations

A tetrahedron is a 3-simplex.

Unlike in the case of other Platonic solids, all vertices are equidistant from each other (they are in the only possible arrangement of four equidistant points).

Tetrahedra are a special type of triangular pyramid and are self-dual. Canonical coordinates of the tetrahedron are (1, 1, 1), (−1, −1, 1), (−1, 1, −1) and (1, −1, −1). A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. The volume of this tetrahedron is 1/3 the volume of the cube. Taking both tetrahedra within a single cube gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).

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

Regular tetrahedra can't tile space by themselves, although it seems likely enough that Aristotle reported it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron which can tile space. This is one of the five Andreini tessellations, and is a limiting case of another, a tiling involving tetrahedra and truncated tetrahedra.

However, there is at least one irregular tetrahedron of which copies can tile space. If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in various 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.)

### Related polyhedra

File:Truncatedtetrahedron.jpg
Truncated tetrahedron
File:Stella octangula.png
Two tetrahedra in a cube

## Intersecting tetrahedra

An interesting polyhedron can be constructed from five intersecting tetrahedra. This compound of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms which are mirror images of each other.

### The isometries of the regular tetrahedron

File:Symmetries of the tetrahedron.png
The proper rotations and reflections in the symmetry group of the regular tetrahedron

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation, showing one of the two tetrahedra in the cube). The symmetries of a regular tetrahedron correspond to half of those of a cube: those which map the tetrahedrons to themselves, and not to each other.

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

The regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to S4. They can be categorized as follows:

• T, isomorphic to alternating group A4 (the identity and 11 proper rotations) with the following conjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the unit quaternion representation):
• identity (identity; 1)
• rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)
• rotation by an angle of 180° such that an edge maps to the opposite edge: 3 ((1 2)(3 4), etc.; i,j,k)
• reflections in a plane perpendicular to an edge: 6
• reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (x is mapped to −x): the rotations correspond to those of the cube about face-to-face axes

## The isometries of irregular tetrahedra

An irregular tetrahedron (3-sided pyramid) with equilateral base and the top above the center has 6 isometries, like an equilateral triangle. In other cases there is no rotational symmetry and at most one mirror plane.

## Computational uses

Complex shapes are often broken down into a mesh of irregular tetrahedra in preparation for finite element analysis and computational fluid dynamics studies.

## Trivia

If each edge of a tetrahedron were to be replaced by a one ohm resistor, the resistance between any two vertices would be 1/2 ohm.

Especially in roleplaying, this solid is known as a d4, one of the more common Polyhedral dice.