A zonotope can also be defined as a convex polytope whose facets are all zonotopes and which has central inversion symmetry.
Properties[edit | edit source]
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).
Examples[edit | edit source]
2D (zonogons)[edit | edit source]
All polygons 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)[edit | edit source]
The prisms of 2D zonotopes are zonotopes as well.
4D (zonochora)[edit | edit source]
Of the uniform polychora, the tesseract, great prismatodecachoron, great disprismatotesseractihexadecachoron, truncated icositetrachoron, great prismatotetracontoctachoron, and great disprismatohexacosihecatonicosachoron are zonotopes.
Duoprisms of two 2n-gons are also zonotopes, as are the prisms of 3D zonotopes.
Zonohedrification[edit | edit source]
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 zonotope. Parallel vectors are ignored, leaving only one in each direction.
Note that a polytope does not have to be a zonotope to be zonohedrified.
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