  We combine the technology of the theory of polytopes and twisted intersection
theory to derive a large class of double copy relations that generalize the
classical relations due to Kawai, Lewellen and Tye (KLT) . To do this, we first
study a generalization of the scattering equations of Cachazo, He and Yuan.
While the scattering equations were defined on $\mathcal{M}_{0,n}$ - the moduli
space of marked Riemann spheres - the new scattering equations are defined on
polytopes known as accordiohedra, realized as hyperplane arrangements. These
polytopes encode as patterns of intersection the scattering amplitudes of
generic scalar theories. The twisted period relations of such intersection
numbers provide a vast generalization of the KLT relations. Differential forms
dual to the bounded chambers of the hyperplane arrangements furnish a natural
generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which
can be determined by counting the number of solutions of the generalized
scattering equations. In this work the focus is on a generalization of the BCJ
expansion to generic scalar theories, although we use the labels KLT and BCJ
interchangeably.
