This

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 t ...

is a symmetric arrangement of 12

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, considered as antiprisms. It can be constructed by superimposing six identical copies of the

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 depicte ...

, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each

is rotated by an equal (and opposite, within a pair) angle θ. Equivalently, a

may be inscribed within each

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

in the compound of six cubes with rotational freedom, which has the same vertices as this compound. When ''θ'' = 0, all six

coincide. When ''θ'' is 45 degrees, the

coincide in pairs yielding (two superimposed copies of) the

compound of six tetrahedra The compound of six tetrahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating ...

. Except if θ = 0 or θ = 45, the compound of twelve tetrahedra with rotational freedom will have a

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of a nonuniform truncated cuboctahedron

Gallery

File:Stellated octahedron (full).stl, ''θ'' = 0° File:Compound of twelve tetrahedra with rotational freedom (5°).stl, ''θ'' = 5° File:Compound of twelve tetrahedra with rotational freedom (10°).stl, ''θ'' = 10° File:Compound of twelve tetrahedra with rotational freedom (15°).stl, ''θ'' = 15° File:Compound of twelve tetrahedra with rotational freedom (20°).stl, ''θ'' = 20° File:Compound of twelve tetrahedra with rotational freedom (25°).stl, ''θ'' = 25° File:Compound of twelve tetrahedra with rotational freedom (30°).stl, ''θ'' = 30° File:Compound of twelve tetrahedra with rotational freedom (35°).stl, ''θ'' = 35° File:Compound of twelve tetrahedra with rotational freedom (40°).stl, ''θ'' = 40° File:Compound of six tetrahedra.stl, ''θ'' = 45°

References

*. Polyhedral compounds {{polyhedron-stub