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Zonotopes are studied from the point of view of central symmetry and howvolumes of facets and the angles between them determine a zonotope uniquely.New proofs are given for theorems of Shephard and McMullen characterizing azonotope by the central symmetry of faces of a fixed dimension. When a zonotopeis regarded as the Minkowski sum of line segments determined by the columns ofa defining matrix, the product of the transpose of that matrix and the matrixacts as a shape matrix containing information about the edges of the zonotopeand the angles between them. Congruence between zonotopes is determined byequality of shape matrices. This condition is used, together with volumecomputations for zonotopes and their facets, to obtain results about rigidityand about the uniqueness of a zonotope given arbitrary normal-vector andfacet-volume data. These provide direct proofs in the case of zonotopes of moregeneral theorems of Alexandrov on the rigidity of convex polytopes, andMinkowski on the uniqueness of convex polytopes given certain normal-vector andfacet-volume data. For a zonotope, this information is encoded in thenext-to-highest exterior power of the defining matrix.

Congruence and Metrical Invariants of Zonotopes