  We find that the equation of $E_8$-singularity possesses two distinct
symmetry groups and modular parametrizations. One is the classical icosahedral
equation with icosahedral symmetry, the associated modular forms are theta
constants of order five. The other is given by the group $\text{PSL}(2, 13)$,
the associated modular forms are theta constants of order $13$. As a
consequence, we show that $E_8$ is not uniquely determined by the icosahedron.
This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious
relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, it
gives a counterexample to Arnold's $A, D, E$ problem, and this also solves the
other related problem on the relation between simple Lie algebras and Platonic
solids. Moreover, we give modular parametrizations for the exceptional
singularities $Q_{18}$, $E_{20}$ and $x^7+x^2 y^3+z^2=0$ by theta constants of
order $13$, the second singularity provides a new analytic construction of
solutions for the Fermat-Catalan conjecture and gives an answer to a problem
dating back to the works of Klein.
