The center points of the equal circles, projected to the sphere surface, constitute the set of points whose minimum distance is maximal.  There are as many minimal edges in the convex hull of this set as their are circle contacts. A few circles may be loose and not touching their nearest neighbors.  But the vast majority are rigid.  If m is the number of rigid circles there will be at least 2m-2 minimal edges in the convex hull.  This provides enough equations to solve for the coordinates of the non-loose points.  The equations to be solved, coming from the distance formula, are polynomial.  It follows (when a solution of the equations exists) that the coordinates of the points (given suitable placement of the polyhedron i.e. convex hull) can all be taken as algebraic numbers.  Since u = cosine( central angle to minimal edge ) is just the dot product of unit vectors to the endpoints of the edge, it is an algebraic expression in the coordinates of those endpoints.  Hence u must be algebraic also.

back to minimal polynomials page