  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
$3 \{3 \} 3 \{4 \} 2$.

Number Theory;Algebraic Geometry