Brian Hayes writes in the recent issue of American Scientist on An Adventure in the Nth Dimension. He beautifully explores the surprising development of the volume (and the surface) of the unit ball as the dimension increases. This is best illustrated by reproducing the graphs of the two functions (extending the domain to the positive reals). The following SAGE code (re)produces the plots which were presented by Paul Renno Heyl (Properties of the locus r = constant in space of n dimensions, 1897).
def V(n):
return pi.n()^(n/2.)/gamma(n/2.+1)
def A(n):
return n*V(n)
G = plot(A, 0, 20, color = 'red', legend_label='surface of unit ball') + list_plot([A(n) for n in srange(20)], color = 'red')
G += plot(V, 0, 20, color = 'blue', legend_label='volume of unit ball') + list_plot([V(n) for n in srange(20)], color='blue')
G.show()
be an 
Hermitian matrix with eigenvalues 
and let the matrix 
be obtained from 
th row and the 
. Then the eigenvalues of 
satisfy the interlacing
.
