“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 basic eigen-analysis of the graph’s adjacency matrix. (Interestingly, while there’s such a thing as spectral graph theory for the study of graph properties via adjacency matrix properties, this particular avenue seems not to have been explored.) The paper also begins the process of cataloging the hundreds of realizations of the uniform polyhedra and their duals. (I really need to construct a proper database of this material.) A future paper will describe how these realizations makes up an additive basis of *all* realizations of a graph.

The Open Question in this:

Is there a nice, mechanical way to generate “good” coordinates for various realizations?

For the figures in the paper, I simply had Mathematica churn out various orthonormal bases for things, and compute coordinates from those, which gives the realizations somewhat random orientations. I don’t doubt that I’m missing-out on some key appearances by the golden ratio and such.