  We introduce a class of polytopes that concisely capture the structure of UV
and IR divergences of general Feynman integrals in Schwinger parameter space,
treating them in a unified way as worldline segments shrinking and expanding at
different relative rates. While these polytopes conventionally arise as convex
hulls - via Newton polytopes of Symanzik polynomials - we show that they also
have a remarkably simple dual description as cut out by linear inequalities
defining the facets. It is this dual definition that makes it possible to
transparently understand and efficiently compute leading UV and IR divergences
for any Feynman integral. In the case of the UV, this provides a transparent
geometric understanding of the familiar nested and overlapping divergences. In
the IR, the polytope exposes a new perspective on soft/collinear singularities
and their intricate generalizations. Tropical geometry furnishes a simple
framework for calculating the leading UV/IR divergences of any Feynman
integral, associating them with the volumes of certain dual cones. As concrete
applications, we generalize Weinberg's theorem to include a characterization of
IR divergences, and classify space-time dimensions in which general IR
divergences (logarithmic as well as power-law) can occur. We also compute the
leading IR divergence of rectangular fishnet diagrams at all loop orders, which
turn out to have a surprisingly simple combinatorial description.
