How to discretize the differential forms on the interval

We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the noncommutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. C-infinity algebra structure. Our main interest lies in a natural `discretization' Cinfinity quasi-isomorphism phi from differential forms to Whitney forms. We establish a uniqueness result that implies that phi coincides with the morphism from homotopy transfer, and obtain several explicit formulas for phi, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Plebanski.