Editing Tetrahedron (section)

===Geometric relations===
A tetrahedron is a 3-[[simplex]]. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space, for an example in [[electromagnetism]] cf. [[Thomson problem]]).

The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, five is the minimum number of tetrahedra required to compose a cube. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be added to make a cube, which has 8 vertices.

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

Regular tetrahedra cannot [[Honeycomb (geometry)|tessellate space]] by themselves, although this result seems likely enough that [[Aristotle]] claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a [[rhombohedron]] that can tile space as the [[tetrahedral-octahedral honeycomb]].

On otherhand, several irregular tetrahedra are known, of which copies can tile space, for instance the [[#Orthoschemes|characteristic orthoscheme of the cube]] and the [[#Disphenoid|disphenoid]] of the [[disphenoid tetrahedral honeycomb]]. The complete list remains an open problem.<ref>{{Cite journal
 | doi = 10.2307/2689983
 | last = Senechal | first = Marjorie | author-link = Marjorie Senechal
 | title = Which tetrahedra fill space?
 | year = 1981
 | journal = [[Mathematics Magazine]]
 | volume = 54
 | issue = 5
 | pages = 227–243
 | publisher = Mathematical Association of America
 | jstor = 2689983
 }}</ref>

If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different 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.)

The tetrahedron is unique among the [[uniform polyhedron|uniform polyhedra]] in possessing no parallel faces.