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 […]

## Archive for the ‘**Harmonious Figures**’ Category

## Extending a Theorem of Barlotti

## Spectral Realizations of Graphs

“Spectral realizations” of a (combinatorial) graph have two important properties: they are harmonious (each graph automorphism induces a rigid symmetry) and eigenic (replacing each vertex with the vector sum of its neighbors yields the same result as scaling the figure). My paper “Spectral Realizations of Graphs” describes a straightforward way of generating the spectral realizations of any graph, using […]

## About the “Harmonious Figures” Category

These articles derive from work I did for my masters thesis on “affinely-regular” figures, which are things like regular polygons and Platonic solids after the application of an affine (or just linear) transformation. The primary result there was that one can decompose any figure (say, a polyhedron) into a “sum” of affinely-regular versions of that […]