Zonotope


A zonotope is a convex polytope that can described as the Minkowski sum of a finite set of line segments in 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 in forms a type of convex polytope called a zonotope. More precisely, a zonotope generated by the vectors is a translation of
where is the matrix whose j'th column is.
The latter description makes it clear that a zonotope is precisely the translation of a projection of an n-dimensional cube.
In the special case where are linearly independent, the zonotope is a parallelotope.
The facets of any zonotope are themselves zonotopes of one lower dimension. Examples of four-dimensional zonotopes include the tesseract, the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope.

Zonotopes and matroids

Fix a zonotope generated by the vectors and let be the matrix whose columns are the. Then the vector matroid on the columns of encodes a wealth of information about, that is, many properties of are purely combinatorial in nature.
For example, pairs of opposite facets of are naturally indexed by the cocircuits of and if we consider the oriented matroid represented by, then we obtain a bijection between facets of and signed cocircuits of which extends to a poset anti-isomorphism between the face lattice of and the covectors of ordered by component-wise extension of. In particular, if and are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment is a zonotope and is generated by both and by whose corresponding matrices, and, do not differ by a projective transformation.

Tilings

Tiling properties of the zonotope are also closely related to the oriented matroid associated to it. First we consider the space-tiling property. The zonotope is said to tile if there is a set of vectors such that the union of all translates is and any two translates intersect in a face of each. Such a zonotope is called a space-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen: The zonotope generated by the vectors 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.
This yields another family of tilings associated to the zonotope, given by a zonotopal tiling of, i.e., a polyhedral complex with support : the union of all zonotopes in the collection is and any two intersect in a common face of each. The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope and single-element lifts of the oriented matroid associated to.

Volume

Zonotopes admit a simple analytic formula for their volume.
Let be the zonotope generated by a set of vectors. Then the d-dimensional volume of is given by
The determinant in this formula makes sense because when the set has cardinality equal to the dimension of the ambient space, the zonotope is a parallelotope.