In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
In geometry, a tetrahedron, 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 ordinary convex polyhedra.
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula, a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's De Divina Proportione, 1509.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
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 each octahedron in the compound of three octahedra.
The compound of four octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by 60 degrees about a separate axis.
In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.
Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry. It arises from superimposing two copies of the corresponding prismatic compound of antiprisms, and rotating each copy by an equal and opposite angle.
Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of prisms sharing a common axis of rotational symmetry. It arises from superimposing two copies of the corresponding prismatic compound of prisms, and rotating each copy by an equal and opposite angle.
This uniform polyhedron compound is a symmetric arrangement of 6 cubes, considered as square prisms. It can be constructed by superimposing six identical cubes, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each cube is rotated by an equal angle θ.
The compound of six tetrahedra with rotational freedom is a uniform polyhedron compound made of a symmetric arrangement of 6 tetrahedra, considered as antiprisms. It can be constructed by superimposing six tetrahedra within a cube, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each tetrahedron is rotated by an equal angle θ. Equivalently, a tetrahedron may be inscribed within each cube in the compound of six cubes with rotational freedom, in such a way as to preserve tetrahedral symmetry.
The compound of eight octahedra with rotational freedom is a uniform polyhedron compound. It is composed of a symmetric arrangement of 8 octahedra, considered as triangular antiprisms. It can be constructed by superimposing eight identical octahedra, and then rotating them in pairs about the four axes that pass through the centres of two opposite octahedral faces. Each octahedron is rotated by an equal angle θ.
The compound of four octahedra with rotational freedom is a uniform polyhedron compound. It consists in a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by an equal angle θ about a separate axis passing through the centres of two opposite octahedral faces, in such a way as to preserve pyritohedral symmetry.
The compound of twenty octahedra with rotational freedom is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra, considered as triangular antiprisms. It can be constructed by superimposing two copies of the compound of 10 octahedra UC16, and for each resulting pair of octahedra, rotating each octahedron in the pair by an equal and opposite angle θ.
This uniform polyhedron compound is a symmetric arrangement of 12 pentagonal antiprisms. It can be constructed by inscribing one pair of pentagonal antiprisms within an icosahedron, in each of the six possible ways, and then rotating each by an equal and opposite angle θ.
In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.
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 mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.
