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

Reflexions on Mahler: Dessins, Modularity and Gauge Theories