  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 non-commutative dg algebra of simplicial cochains
and iii) the Whitney forms, equipped with a homotopy commutative and homotopy
associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a
natural `discretization' $C_\infty$ quasi-isomorphism $\varphi$ from
differential forms to Whitney forms. We establish a uniqueness result that
implies that $\varphi$ coincides with the morphism from homotopy transfer, and
obtain several explicit formulas for $\varphi$, all of which are related to the
Magnus expansion. In particular, we recover combinatorial formulas for the
Magnus expansion due to Mielnik and Pleba\'nski.
