On Folding a Polygon to a Polyhedron

We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.Comment: 6 pages, 1 figur

Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding

Nonorthogonal Polyhedra Built from Rectangles

We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].Comment: 19 pages, 20 figures. Revised version makes two corrections: The statement of the old Lemma 14 was incorrect. It has been corrected and merged with Lemma 13 now. Second, Figure 19 (a skew quadrilateral) was incorrect, and is now removed. It played no substantive role in the proof

Folding Orthogonal Polyhedra

In this thesis, we study foldings of orthogonal polygons into orthogonal polyhedra. The particular problem examined here is whether a paper cutout of an orthogonal polygon with fold lines indicated folds up into a simple orthogonal polyhedron. The folds are orthogonal and the direction of the fold (upward or downward) is also given. We present a polynomial time algorithm to solve this problem. Next we consider the same problem with the exception that the direction of the folds are not given. We prove that this problem is NP-complete. Once it has been determined that a polygon does fold into a polyhedron, we consider some restrictions on the actual folding process, modelling the case when the polyhedron is constructed from a stiff material such as sheet metal. We show an example of a polygon that cannot be folded into a polyhedron if folds can only be executed one at a time. Removing this restriction, we show another polygon that cannot be folded into a polyhedron using rigid material

Enumerating Foldings and Unfoldings between Polygons and Polytopes

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in Proceedings of the Japan Conference on Discrete and Computational Geometry, Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701