The compound of six tetrahedra is a

uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...

. It's composed of a symmetric arrangement of 6

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

. It can be constructed by inscribing a

stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted ...

within each

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

in the

compound of three cubes In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry. It has been depicted in works by Max Brückner and M.C. Escher. History This compound appears in Max Brückner's ...

, or by stellating each octahedron in the

compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathemati ...

. It is one of only five polyhedral compounds (along with the

compound of two great dodecahedra This uniform polyhedron compound is a composition of 2 great dodecahedra, in the same arrangement as in the compound of 2 icosahedra. It is one of only five polyhedral compounds (along with the compound of six tetrahedra, the compound of five g ...

, the

compound of five great dodecahedra This uniform polyhedron compound is a composition of 5 great dodecahedra, in the same arrangement as in the compound of 5 icosahedra. It is one of only five polyhedral compounds (along with the compound of six tetrahedra, the compound of two g ...

, the

compound of two small stellated dodecahedra This uniform polyhedron compound is a composition of 2 small stellated dodecahedra, in the same arrangement as in the compound of 2 icosahedra. It is one of only five polyhedral compounds (along with the compound of six tetrahedra, the compou ...

, and the

compound of five small stellated dodecahedra This uniform polyhedron compound is a composition of 5 small stellated dodecahedra, in the same arrangement as in the compound of 5 icosahedra. It is one of only five polyhedral compounds (along with the compound of six tetrahedra, the compoun ...

) which is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

and

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

but not

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

References

*. Polyhedral compounds {{polyhedron-stub