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Tetrahedron

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.

A tetrahedron is a 3-simplex.

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 tetrahedron can be embedded inside a cube so that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. Taking both tetrahedra within a single cube gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. 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. In fact, octahedra are necessary to fill some of the gaps. This is one of the five Andreini tessellations, and is a limiting case of another, a tiling involving tetrahedra and truncated tetrahedra.

The volume of an irregular tetrahedron, given its verticies A, B, C and D, is |det(A-B, B-C, C-D)|, or any other combination of pairs of verticies that form a simply connected graph. (This works for regular tetrahedrons too.)

Like all platonic solids, archimedean solids and a lot (maybe all, not sure) of convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

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