This is a long, complex, and incredibly rich paper. It contains, in some sense, one main result: the source unfolding that unfolds the surface of a convex polyhedron P to a planar, nonoverlapping polygon that generalizes to higher dimensions.

This unfolding is defined with respect to a point x on the polyhedron P. This point determines the cut locus K_x: roughly, the set of all the endpoints of shortest paths emanating from x and covering the surface of P. The idea is that one “cuts” along the network of lines that constitute the cut locus, and the surface unfolds like a flower to the planar polygon , and notably without overlapping itself.

The generalization to higher dimensions starts with a convex polytope P in dimension d, and establishes the analogous result: “cutting” the cut locus K_x for a point x results in a nonoverlapping “polyhedral foldout” in dimension d-1. The cutting is now a higher-dimensional slice. For example, if P is the hypercube in d=4, and x a point in the center of a cubical facet, then the source unfolding is a polyhedron in 3-space with six “spikes” symmetrically surrounding a central cube.

One of the authors’ achievements en route to their main result is precisely defining the cut locus in the higher-dimensional context and establishing its properties. Another achievement is an algorithm for constructing the polyhedral foldout.

The paper closes with a series of observations concerning possible extensions, and a series of conjectures that map out many fascinating lines of research. The authors’ last conjecture--that self-intersection can be avoided during the “blooming” process as well as in the final foldout--has recently been settled affirmatively for three-dimensional polyhedra [1], but remains untouched for d > 3.