The associahedra is a sequence of polytopes that controls higher associative algebraic structures. Its concreteness allows explicit combinatorial manipulation, e.g. a (chain complex level) definition of A_\infty algebras.

I will explain “the magical formula” for the diagonal map of the associahedra that gives a computable description of the tensor product of two A_\infty algebras.

It is derived from a general construction, motivated by the theory of fiber polytopes, of a "polytopal" approximation for the diagonal of any polytope. For simplices, it recovers the classical Alexander-Whitney maps.

Bonus: compatibility with the diagonal construction forces a unique polytopal operad structure on associahedra, which (surprisingly) has never been given explicitly. This is joint work with Thomas, Tonks, and Vallette.