I will address the string theory part of the question. String theory describes natural bridges between Kleinian singularities (and therefore Platonic solids), ALE spaces, quiver diagrams, ADE diagrams and two dimensional Conformal Field Theories. The scene is given by compactifications of string theory on Kleinian orbifolds $M_\Gamma=\mathbb{C}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $SU(2)$. The space $M_\Gamma$ admits a Kleinian singularity at the origin. After studying this string theory system, one is less surprised to see that all these different structures (Kleinian singularities, quiver diagrams, ALE spaces, ADE diagrams, 2D Conformal Field Theories) admit the same ADE classifications since they provide different descriptions of the same underlying physical system.
Michael Douglas and Gregory Moore have studied the compactification of string theory on Kleinian orbifold $M_\Gamma$ using D-branes as probes of the geometry. D-branes provide a physical description of the geometry in terms of supersymmetric gauge theories. Such supersymmetric gauge theories are efficiently summarized by a tamed quiver diagram with a very natural physical interpretation.
The vacuum of these supersymmetric gauge theories depends on a potential whose construction is equivalent to the hyperkhaler quotient construction of Asymptotic Locally Euclidian Spaces (ALE spaces) given by Kronheimer. ALE spaces are HyperKahler four dimensional real manifolds whose anti-self-dual metrics are asymptotic to a Kleinian orbifold $M_\Gamma=\mathbb{C}^4/ \Gamma$. Physically ALE spaces described gravitational instantons. ALE spaces provide small resolutions of the Kleinian singularities where the singular point is replaced by a system of spheres whose intersection matrix is equivalent to the Cartan matrix of an ADE Dynkin diagram. One can also consider Yang-Mills instantons on such spaces. The gauge group associated with the Yang-Mills instantons is given by the type of ADE diagram obtained by the resolution of the singularity. This was analyzed in the math literature by Kronheimer and Nakajima.
Finally, the link between D-branes on ALE spaces (or equivalently Kleinian singularities) and the ADE classification of two dimensional Conformal Field Theories (CFT) was studied by Lershe, Lutken and Schweigert. In the description of the CFT, one recovers Arnold's ADE list of simple isolated singularities.