  In this paper, we will describe the space spanned by the angle-sums of
polytopes, recorded in the alpha-vector. We will consider the angles sums of
simplices and the angles sums and face numbers of simplicial polytopes and
general polytopes. We will construct families of polytopes whose angle sums
span the spaces of polytopes defined by the Gram and Perles equations,
analogues of the Euler and Dehn-Sommerville equations. We show that the
dimensions of the affine span of the space of angle sums of simplices is
floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial
polytopes and general polytopes are d-1 and 2d-3 respectively.
