Starting from comparably explicit objects (algebraic cycles), Bloch and Kriz have given a tentative definition of a small yet rich category of motives (mixed Tate motives), at least over a field. They also exhibited a distinguished class of cycles corresponding to polylogarithms. One can also find _multiple_ polylogarithms as algebraic cycles, and it turns out that their differential structure can be conveniently described with the help of combinatorics of polygons. This leads to a coproduct on polygons which is a variant of the Connes-Kreimer coproduct on rooted trees.