  We derive the structure of the Higgs branch of 5d superconformal field
theories or gauge theories from their realization as a generalized toric
polygon (or dot diagram). This approach is motivated by a dual, tropical curve
decomposition of the $(p,q)$ 5-brane-web system. We define an edge coloring,
which provides a decomposition of the generalized toric polygon into a refined
Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The
Coulomb branch of the magnetic quiver is then conjecturally identified with the
5d Higgs branch. Furthermore, from partial resolutions, we identify the
symplectic leaves of the Higgs branch and thereby the entire foliation
structure. In the case of strictly toric polygons, this approach reduces to the
description of deformations of the Calabi-Yau singularities in terms of
Minkowski sums.
