Abstract
The authors study the fundamental properties of scattering amplitudes of particles in any spacetime dimension. They introduce binary geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. Furthermore, they define open and closed ``cluster string integrals'', which provide a generalization of particle and string scattering amplitudes, and enjoy remarkable factorization properties at finite ${\ensuremath{\alpha}}^{\ensuremath{'}}$.
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