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 *d*x*d* 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.