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 figure: a random quadrilateral is the “sum” of parallelograms. Even better, it turns out that one can decompose an affinely-regular figure into a “sum” of actually-regular versions. (See my post on the Theorem of Barlotti.) Ultimately, these two results imply that any figure is the “sum” of regular versions of itself: squash and stretch and distort a cube any way you like (while keeping the edges straight), and it’ll be the sum of un-distorted cubes of various sizes and orientations. Very cool!

“Harmony” is a term that generalizes the basic notion of regularity. An harmonious realization of a (combinatorial) graph is one in which all automorphisms of the graph are induced by isometries of the realization. That is, an harmonious realization is the best-possible way to draw the graph, since it shows all the automorphic structure; this can be helpful for visualization purposes … except when the realizations exist in more dimensions than we can see. Generating harmonious realizations turns out to be a pretty straightforward —though not-well-known— exercise in the eigen-analysis of a graph’s adjacency matrix. (See my post on “spectral” realizations … if only for the pictures.)