Tetrahedron

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A tetrahedron is a Platonic solid composed of four triangular faces, with three meeting at each vertex. Tetrahedra are a special type of triangular pyramid and are self-dual. 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.