Tetrahedron

The tetrahedron, or tet 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.

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


 * ($\sqrt{6}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4).

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:


 * (±1/2, –$\sqrt{2}$/6, –$\sqrt{2}$/12),
 * (0, $\sqrt{3}$/3, –$\sqrt{6}$/12),
 * (0, 0, $\sqrt{3}$/4).

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

Related polyhedra
Two tetrahedra cn be attached at a common face to form a triangular bipyramid, 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.