Closed Formula for the Number of Parts Great Spheres Divide an N-Sphere

This paper derives a recursive relationship and closed formula for the number of pieces of dimensional sphere when divided by great spheres in general position. There are two main parts of this paper. The first is the proof of the recursive relationship and closed formula of a possibly known problem: the number of pieces separated by copies of -dimensional planes in general position in a -dimensional Euclidean space. The second part is inspired by the first part, deriving a recursive relationship and closed formula for a similar problem but in spherical Geometry: the number of pieces separated by copies of -dimensional great spheres in general position in a -dimensional sphere. As we will explain, the sphere problem we solve does not reduce to the planes problems we discuss first.