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 tran ...
is a symmetric arrangement of 12
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 ...
, considered as
antiprisms
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass ...
. 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 depicted ...
, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each
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 ...
is rotated by an equal (and opposite, within a pair) angle θ. Equivalently, 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 ...
may be inscribed 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 six cubes with rotational freedom, which has the same vertices as this compound.
When ''θ'' = 0, all six
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 ...
coincide. When ''θ'' is 45 degrees, 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 depicted ...
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 stellatin ...
.
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