  The values of the determinant of Vandermonde matrices with real elements are
analyzed both visually and analytically over the unit sphere in various
dimensions. For three dimensions some generalized Vandermonde matrices are
analyzed visually. The extreme points of the ordinary Vandermonde determinant
on finite-dimensional unit spheres are given as the roots of rescaled Hermite
polynomials and a recursion relation is provided for the polynomial
coefficients. Analytical expressions for these roots are also given for
dimension three to seven. A transformation of the optimization problem is
provided and some relations between the ordinary and generalized Vandermonde
matrices involving limits are discussed.
