6.6CGJul 10
Overlapping Unfoldings of Cones and Convex PolyhedraMIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine et al. · mit
Research on Dürer's problem focuses on edge unfoldings of convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t. We have two main results. The first is that, if we allow unfolding cuts that do not follow polyhedron edges, then there is a convex polyhedron that can unfold with overlap of any given thickness. The second result is that for any given thickness, there is a convex polyhedron with an edge unfolding that overlaps to that thickness.
7.9CGMar 10
Prismatoid Band-Unfolding RevisitedJoseph O'Rourke
It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.