A zonotope is a convex polytope that can be described as the Minkowski sum of a finite set of line segments in <math>\mathbb{R}^d</math> 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 <math>\mathbb{R}^d</math> forms a type of convex polytope called a zonotope. More precisely, a zonotope <math>Z</math> generated by the vectors <math>w_1,...,w_n\in\mathbb{R}^d</math> is a translation of
<math display=block>
Z = \{a_1 w_1 + \cdots + a_n w_n | \; 0 \le a_j \le 1 \text{ for all } j \}
= \mathbf{W} \, [0,1]^n ,
</math>
where <math>\mathbf{W}</math> is the <math>d \times n</math> matrix whose jth column is <math>w_j</math>.
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 <math>w_1,...,w_n\in\mathbb{R}^d</math> are linearly independent, the zonotope <math>Z</math> 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
Fix a zonotope <math>Z</math> generated by the vectors <math>w_1,\dots,w_n \in \mathbb{R}^d</math> and let <math>\mathbf{W}</math> be the <math>d \times n</math> matrix whose columns are the <math>w_i</math>. Then the vector matroid <math>{\mathcal{M</math> on the columns of <math>\mathbf{W}</math> encodes a wealth of information about <math>Z</math>, that is, many properties of <math>Z</math> are purely combinatorial in nature.
For example, pairs of opposite facets of <math>Z</math> are naturally indexed by the cocircuits of <math>\mathcal{M}</math> and if we consider the oriented matroid <math>\mathcal{M}</math> represented by <math>\mathbf{W}</math>, then we obtain a bijection between facets of <math>Z</math> and signed cocircuits of <math>\mathcal{M}</math> which extends to a poset anti-isomorphism between the face lattice of <math>Z</math> and the covectors of <math>\mathcal{M}</math> ordered by component-wise extension of <math>0 \prec +, -</math>. In particular, if <math>M</math> and <math>N</math> 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 <math>[0,2] \subset \mathbb{R}</math> is a zonotope and is generated by both <math>\{2\mathbf{e}_1\}</math> and by <math>\{\mathbf{e}_1, \mathbf{e}_1\}</math> whose corresponding matrices, <math>[2]</math> and <math>[1~1]</math>, do not differ by a projective transformation.
Tilings
Tiling properties of the zonotope <math>Z</math> are also closely related to the oriented matroid <math>\mathcal{M}</math> associated to it. First we consider the space-tiling property. The zonotope <math>Z</math> is said to tile <math>\mathbb{R}^d</math> if there is a set of vectors <math>\Lambda \subset \mathbb{R}^d</math> such that the union of all translates <math>Z + \lambda</math> (<math>\lambda \in \Lambda</math>) is <math>\mathbb{R}^d</math> 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: The zonotope <math>Z</math> generated by the vectors <math>V</math> 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 <math>Z</math>, given by a zonotopal tiling of <math>Z</math>, i.e., a polyhedral complex with support <math>Z</math>: the union of all zonotopes in the collection is <math>Z</math> and any two intersect in a common (possibly empty) face of each. The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope <math>Z</math> and single-element lifts of the oriented matroid <math>\mathcal{M}</math> associated to <math>Z</math>.
Volume
Zonotopes admit a simple analytic formula for their volume.
Let <math>Z(S)</math> be the zonotope <math>Z = \{a_1 w_1 + \cdots + a_n w_n | \; \forall(j) a_j\in [0,1]\}</math> generated by a set of vectors <math>S = \{w_1,\dots,w_n\in\mathbb{R}^d\}</math>. Then the d-dimensional volume of <math>Z(S)</math> is given by
:<math>\sum_{T\subset S \; : \; |T| = d} |\det(Z(T))|</math>
The determinant in this formula makes sense because (as noted above) when the set <math>T</math> has cardinality equal to the dimension <math>n</math> of the ambient space, the zonotope is a parallelotope.