We propose a computational, convex hull free framework that takes advantage of the combinatorial structure of a zonotope, as for example its symmetry group, to orbitwise generate all canonical representatives of its vertices. We illustrate the proposed framework by generating the 1 955 230 985 997 140 vertices of the 9-dimensional White Whale. We also compute the number of edges of this zonotope up to dimension 9 and exhibit a family of vertices whose degree is exponential in the dimension. The White Whale is the Minkowski sum of the \(2^d-1\) non-zero 0/1-valued d-dimensional vectors. The central hyperplane arrangement dual to the White Whale, made of the hyperplanes normal to these vectors, is the resonance arrangement and has been studied in various contexts including algebraic geometry, mathematical physics, economics, psychometrics, and representation theory.