Tetrahedron

The tetrahedron or tet, also sometimes called the 3-simplex, is the simplest possible non-degenerate polyhedron. The full symmetry version has 4 equilateral triangles as faces, joining 3 to a vertex, and is one of the 5 Platonic solids. It is the 3-dimensional simplex.

It is the uniform digonal antiprism and regular-faced triangular pyramid. Both of these forms are convex segmentohedra.

A regular tetrahedron of edge length $\sqrt{2}$ can be inscribed in the unit cube. In fact the tetrahedron is the alternated cube, which makes it the 3D demihypercube. The next larger simplex that can be inscribed in a hypercube is the octaexon.

The tetrahedron occurs as cells of three of the six convex regular polychora, namely the pentachoron, hexadecachoron, and hexacosichoron.

Vertex coordinates
Coordinates for the vertices of a tetrahedron can be given by all even changes of sign of:


 * $$\left(\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4}\right).$$

These arise from the fact that a tetrahedron can be constructed as the alternation of the cube.

Alternate coordinates can be derived from those of the triangle, by considering the tetrahedron as a triangular pyramid:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4}\right).$$

These are more complicated, but generalize to simplexes of any dimension.

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


 * $$\left(\frac{\sqrt{2}}{2},\,0,\,0,\,0\right).$$

Representations
A regular tetrahedron can be represented by the following Coxeter diagrams:


 * x3o3o (full symmetry)
 * s2s4o (digonal antiprism, is generally a tetragonal disphenoid)
 * s2s2s (alternated cuboid, generally a rhombic disphenoid)
 * ox3oo&#x (A2 axial, generally a triangular pyramid)
 * xo ox&#x (A1×A1 axial, generally a digonal disphenoid)
 * oox&#x (A1 only, generally a sphenoid)
 * oooo&#x (no symmetry, fully irregular tetrahedron)

Related polyhedra
Two tetrahedra can be attached at a common face to form a triangular tegum, one of the Johnson solids.

A tetrahedron can also be elongated by attaching a triangular prism to one of the faces, forming the elongated triangular pyramid.

A number of uniform polyhedron compounds are composed of tetrahedra:


 * Stella octangula (2)
 * Chiricosahedron (5)
 * Icosicosahedron (10)
 * Snubahedron (6)
 * Small snubahedron (6, with rotational freedom)
 * Disnubahedron (12, with rotational freedom)
 * An infinite number of prismatic compounds that are antiprisms of compounds of digons (where the digons degenerate to edges).

Other kinds of tetrahedra
Besides the regular tetrahedron, there are a number of other polyhedra containing four triangular faces. Tetrahedra are generally classified by symmetry. Some of these classes of tetrahedra include:


 * Triangular pyramid - one equilateral triangle (base) and three identical isosceles triangles
 * Tetragonal disphenoid - four identical isosceles triangles
 * Digonal disphenoid - Two pairs of identical isosceles triangles
 * Rhombic disphenoid - Four identical scalene triangles
 * Phyllic disphenoid - Two pairs of identical scalene triangles
 * Sphenoid - Only a single symmetry axis
 * Irregular tetrahedron - No symmetry axes at all