Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

It is well-known that Klein's lectures on the icosahedron and the solution of equations of fifth degree is one of the most important and influential books of 19th-century mathematics. In the present paper, we will give the complex counterpart of Klein's book, i.e., a story about complex regular polyhedra. We will show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group ${\frak G}^{\prime}_{216}$. There are two finite unitary groups generated by reflections corresponding to this collineation group. One, of order 648, is the symmetry group of the complex regular polyhedron $3 \{3 \} 3 \{3 \} 3$, and the other, of order 1296, is the symmetry group of the regular complex polyhedron $2 \{4 \} 3 \{3 \} 3$ or its reciprocal $3 \{3 \} 3 \{4 \} 2$.