We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come… (More)
A geometric graph G is a graph whose vertex set is a set Pn of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this… (More)
We show that every convex polyhedron may be unfolded to one pla-nar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are " edge-unfold rigid " in the sense that each of their unfoldings may only fold back to the original. For example, each of the… (More)
We give a construction of a uniform covering of 2-paths with 5-paths in K n for all even n ≥ 6, i.e., we construct a set S of 5-paths in K n having the property that each 2-path in K n lies in exactly one 5-path in S for all even n ≥ 6.