How do we extend the 2 -dim'l familiar fact that "the sum of the angles of
a triangle equals \pi " , to hight dim'l polyhedra?
We begin with a natural discretization of the volume of a spherical
polytope, and define certain natural analogues of theta functions,
called cone theta functions. These meromorphic functions help us
analyze the structure of spherical polytopes.
It turns out that we can obtain good asymptotics of cone theta
functions, attached to any simplicial polyhedral cone, near a rational
'cusp', and we use these asymptotics to give new extensions of the
Gram relations for the solid angles of faces of a simple rational
polytope. These new relations involve 'Gauss' sums over
parallelepipeds, and we will introduce all objects from 'almost first
principles', including lots of pictures.