Abstract：Fix a 4-dimensional convex body and consider the areas of its orthogonal projections onto the six coordinate planes. This associates to each convex body a 6-tuple of nonnegative numbers. I will explain a surprising connection: the set of all such 6-tuples coincides with the set of all 4-by-4 symmetric matrices with nonnegative entries, zero diagonal, and Lorentzian signature. And in addition, this set is also equal to the Grassmannian over a certain triangular hyperfield. This talk is based on joint work with Huang, Huh, Michałek, and Wang.