  We provide a unified framework of Mahler measure, dessins d'enfants, and
gauge theory. With certain physically motivated Newton polynomials from
reflexive polygons, the Mahler measure and the dessin are in one-to-one
correspondence. From the Mahler measure, one can construct a Hauptmodul for a
congruence subgroup of the modular group, which contains the subgroup
associated to the dessin. In brane tilings and quiver gauge theories, the
modular Mahler flow gives a natural resolution of the inequivalence amongst the
three different complex structures $\tau_{R,G,B}$. We also study how, in
F-theory, 7-branes and their monodromies arise in the context of dessins.
Moreover, we give a dictionary on how Mahler measure generates Gromov-Witten
invariants.
