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We give an interpretation of a class of discrete-to-continuum results for
Ising systems using the theory of zonoids. We define the classes of rational
zonotopes and zonoids, as those of the Wulff shapes of perimeters obtained as
limits of finite-range homogeneous Ising systems and of general homogeneous
Ising systems, respectively. Thanks to the characterization of zonoids in terms
of measures on the sphere, rational zonotopes, identified as finite sums of
Dirac masses, are dense in the class of all zonoids. Moreover, we show that a
rational zonoid can be obtained from a coercive Ising system if and only if the
corresponding measure satisfies some connectedness properties, while it is
always a continuum limit of discrete Wulff shapes under the only condition that
the support of the measure spans the whole space. Finally, we highlight the
connection with the homogenization of periodic Ising systems and propose a
generalized definition of rational zonotope of order N, which coincides with
the definition of rational zonotope if N=1