Polygonal combinatorics for algebraic cycles

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.