A zonotope is a convex polytope that can be described as the Minkowski sum or, equivalently as a projection of a hypercube. Zonotopes are intimately connected to hyperplane arrangements and matroid theory.

Definition and basic properties

The Minkowski sum of a finite set of line segments forms a type of convex polytope is a translation of

matrix whose j. The latter description makes it clear that a zonotope is precisely the translation of a projection of an n-dimensional cube.

is a (possibly lower-dimensional) parallelotope.

The facets of any zonotope are themselves zonotopes of one lower dimension. Examples of four-dimensional zonotopes include the tesseract (Minkowski sums of mutually perpendicular equal length line segments), the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope.

Zonotopes and matroids

. Then the vector matroid are purely combinatorial in nature.

and if we consider the oriented matroid which extends to a poset anti-isomorphism between the face lattice are two matrices that differ by a projective transformation, do not differ by a projective transformation.

Tilings

is said to tile and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called a space-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen: tiles space if and only if the corresponding oriented matroid is regular. So the seemingly geometric condition of being a space-tiling zonotope actually depends only on the combinatorial structure of the generating vectors.

Dissections

Every d-dimensional zonotope generated by a finite set A of vectors can be partitioned into parallelepipeds, with one parallelepiped for each linearly independent subset of A., given by a zonotopal tiling and single-element lifts.

Volume

Zonotopes admit a simple analytic formula for their volume.

. Then the d is given by

of the ambient space, the zonotope is a parallelotope.

References