We define a Hopf algebra of motivic iterated integrals on the line
and prove an explicit formula for the coproduct $\Delta$ in this
Hopf algebra. We show that this formula encodes the group law of
the automorphism group of a certain noncommutative variety. We
relate the coproduct $\Delta$ to the coproduct in the Hopf algebra
of decorated rooted plane trivalent trees, which is a plane
decorated version of the one defined by Connes and Kreimer
[CK]. As an application, we derive explicit formulas for the
coproduct in the motivic multiple polylogarithm Hopf algebra.
These formulas play a key role in the mysterious correspondence
between the structure of the motivic fundamental group of
$\mathbb{P}^1 - (\{0, \infty\}\cup \mu_N)$ , where $\mu_N$ is the
group of all $N$ th roots of unity, and modular varieties for
${\GL_m}$ (see [G6], [G7]).
In Section 7 we discuss some general principles relating Feynman
integrals and mixed motives. They are suggested by Section 4 and
the Feynman integral approach for multiple polylogarithms on
curves given in [G7]. The appendix contains background
material.