Zonotope

A zonotope is a polytope that is the Minkowski sum of a set of vectors. These vectors are called its generators.

A zonotope can also be defined in other terms:


 * a polytope that can be alternated
 * a polytope whose facet s are all convex and have 180° rotational symmetry ("point symmetry")

Properties
The faces of a zonotope always have an even number of edges, and the facets of a zonotope are themselves zonotopes.

An n-dimensional zonotope can always be dissected into "primitive" zonotopes (which are n-parallelotopes, or the generalization of parallelepipeds to n dimensions).

2D (zonogons)
All polygon s with point symmetry (that is, they have a symmetry that rotates them 180° about the center) are zonotopes, and all convex regular 2n-gons are zonotopes. These are usually found as the faces of higher-dimensional zonotopes.

3D (zonohedra)
Of the uniform polyhedra, the cube and the three omnitruncates (the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron) are zonotopes.

Two of the uniform dual polyhedra - the rhombic dodecahedron and rhombic triacontahedron - are also zonotopes.

Removing one or two generators from the rhombic triacontahedron results in the rhombic icosahedron and Bilinski dodecahedron, respectively. Both are also zonotopes.

The prism s of 2D zonotopes are zonotopes as well.

4D (zonochora)
Of the uniform polychora, the tesseract, great prismatodecachoron, great disprismatotesseractihexadecachoron, truncated icositetrachoron, great prismatotetracontoctachoron, and great disprismatohexacosihecatonicosachoron are zonotopes.

Duoprism s of two 2n-gons are also zonotopes, as are the prisms of 3D zonotopes.

Zonohedrification
Zonohedrification is an operation upon a polytope (like truncation, the dual, or many others) that regards the vertices of the starting polytope as vectors (originating from the polytope's center), then uses those vectors to generate a zonohedron. Parallel vectors are ignored, leaving only one in each direction.

Note that a polytope does not have to be a zonotope to be zonohedrified.