Barlotti’s Theorem states that an “affinely regular” polygon is the vertex sum of two regular polygons of the same type. This result can be interpreted as a straightforward decomposition of 2×2 matrices, which in turn can be extended to a decomposition of dxd matrices, which immediately gives rise to the (Extended) Barlotti Theorem for Multiple Dimensions.
My paper —aptly entitled “An Extension of a Theorem of Barlotti to Multiple Dimensions”— discusses generalizing the notion of vertex sum to “point sum”, so that the Extended Barlotti provides such statements as any ellipsoid is the point sum of three spheres.
The Extended Barlotti Theorem implies that spectral realizations of a graph (see this post) form an additive basis of all realizations of that graph; that is, any realization can be expressed as the vertex sum of (harmonious) spectral realizations.