  In this note we continue the exploration of the polytope picture for
scattering amplitudes, where amplitudes are associated with the volumes of
polytopes in generalized momentum-twistor spaces. After a quick warm-up example
illustrating the essential ideas with the elementary geometry of polygons in
CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in
CP^3x CP^3, which can be thought of as the space obtained by taking the
geometric dual of the Wilson loop in each CP^3 of the product. We then review
the polytope picture for the NMHV tree amplitude and give it a more direct and
intrinsic definition as the geometric dual of a canonical "square" of the
Wilson-Loop polygon, living in a certain extension of momentum-twistor space
into CP^4. In both cases, one natural class of triangulations of the polytope
produces the BCFW/CSW representations of the amplitudes; another class of
triangulations leads to a striking new form, which is both remarkably simple as
well as manifestly cyclic and local.
