Flat Zipper-Unfolding Pairs for Platonic Solids Joseph O Rourke October, 00 Abstract We show that four of the five Platonic solids surfaces ma be cut open with a Hamiltonian path along edges and unfolded to a polgonal net each of which can zipper-refold to a flat doubl covered parallelogram, forming a rather compact representation of the surface. Thus these regular polhedra have particular flat zipper pairs. No such zipper pair eists for a dodecahedron, whose Hamiltonian unfoldings are zip-rigid. This report is primaril an inventor of the possibilities, and raises more questions than it answers. Introduction It has been known since the time of Aleandrov and it was certainl known to him that the surface of a polhedron could sometimes be cut open to a net and refolded to a doubl covered polgon, which we will henceforth call a flat polhedron. Such flat polhedra are eplicitl countenanced in Aleandrov s 9 gluing theorem. Perhaps the first specific eample of this possibilit occurred in [LO9], which included the eample illustrated in Figure : the familiar Latin-cross unfolding of the cube ma be refolded to a flat conve quadrilateral polhedron. This is one of the two flat conve polhedron that ma be folded from the Latin cross [DO07, Fig..]. Let us sa that two polhedra Q and Q form a net pair if the ma be unfolded to a common polgonal net. In Figure, the cube is cut along edges to unfold to the Latin cross polgon, but the flat quadrilateral must have face cuts through the interior of its two faces to unfold to the same Latin cross. In general there is little understanding of which polhedra form net pairs. See, for eample, Open Problem. in [DO07]. Here we eplore a narrow question on net pairs, narrow enough to obtain a complete answer. The cuts to unfold a conve polhedron to a single polgon form a spanning tree of the polhedron s vertices [DO07, Sec...]. Shephard eplored the special case where the spanning tree is a Hamiltonian path of the -skeleton Department of Computer Science, Smith College, Northampton, MA 00, USA. orourke@cs.smith.edu. See [DO07, Sec..] and [Pak0, Sec. 7] for descriptions of this theorem.
Figure : Folding the Latin cross cube net to a flat quadrilateral polhedron. Points with the same label in are identified in the refolding. of the polhedron, i.e., all cuts are along polhedron edges [She7]. The result is a Hamiltonian unfolding of the polhedron. (Note the cube unfolding that produces Figure is not a Hamiltonian unfolding: the cut tree has four leaves.) Some combinatorial questions on Hamiltonian unfoldings were eplored in [DDLO0]; see [DO07, Fig..9 ]. In particular, there are polhedra that have an eponential number of combinatoriall distinct Hamiltonian unfoldings: Ω(n) for a polhedron with n vertices. Another variant is provided b the class of perimeter-halving foldings [DO07, Sec... ], which correspond to spanning cut paths that ma emplo face cuts rather than solel following polhedron edges. In [LDD + 0] these paths were memorabl rechristened as zipper paths, producing zipper unfoldings. We will adopt that nomenclature, including the verbs zip and unzip to mean folding and unfolding (respectivel) along zipper paths. We reserve Hamiltonian path to be a zipper path along polhedron edges. Finall, if two polhedra each unzip to a common polgonal net, we sa the form a zipper pair. The narrow question we eplore is this: Question: Does each of the Platonic solids form a zipper pair with a flat conve polhedron, with the zipper path on the regular polhedron forming a Hamiltonian path of its edges? We show that the tetrahedron, the cube, the octahedron, and the icosahedron all form such zipper pairs with flat parallelogram polhedra. The dodecahedron has no such zipper mate. Note that it would be too restrictive to insist We drop the modifier regular to shorten the names of the five regular polhedra.
that both zippers are Hamiltonian paths of the -skeletons, because for a flat polhedron, the -skeleton is the single ccle bounding the polgon, and so a Hamiltonian unfolding is just two copies of the conve polgon joined along one edge. Flat Zipper Pairs. Tetrahedron The regular tetrahedron has onl one Hamiltonian path (up to smmetries), which unfolds to the parallelogram shown in Figure (in Fig. in [LDD + 0]). Because this net is a conve polgon, Thm... in [DO07] establishes that it has an infinite number of zippings to various conve polhedra. The zipping shown in Figure (c) folds it to a doubl covered rhombus. (c) Figure : The Hamiltonian cut path on a tetrahedron leads to the Hamiltonian unfolding, which zips from to (identifing the labeled points) to a flat rhombus polhedron of side length (c).. Cube The cube has three distinct Hamiltonian unfoldings (Fig. in [LDD + 0]): one with the path endpoints at opposite cube corners, and two with the path endpoints at either end of a cube edge. One of the latter (shown in Figure ) produces a T -shape that has no zippings ecept back to the cube. We call such a zipper unfolding zip-rigid. We defer an eplanation of how it is known that this unfolding is zip-rigid to Section. below.
Figure : The first Hamiltonian cut path leads to a zip-rigid Hamiltonian T -unfolding of the cube. The other two Hamiltonian unfoldings of the cube, which we call the S - and the Z -unfoldings, both zip to the same doubl covered parallelogram, as shown in Figures and. An animation of the S -folding is shown in Figure.. A Zipping Algorithm Let P be a polgon, the polgonal net corresponding to a zip-pair of polhedra Q and Q. Each of the two zippings of P are perimeter-halving foldings, with the endpoints of the zip path bisecting the perimeter. If we normalize the perimeter of P to and parametrize it from 0 to, we can view the two zippings abstractl as in Figure 7. One of the zip-path endpoints are at 0 and, and the other zip-path endpoints are at and = +. We seek to find all the locations that determine a zipping to some conve polhedron. As previousl mentioned, if P is conve, then ever determines a conve polhedron (Thm... in [DO07]), so we henceforth eclude that case. If P is not conve, it has at least one refle verte v with internal angle β > π. Now there are onl two options at v: () v can serve as, so the zipping starts at v = ; or () Some strictl conve verte u i whose internal angle α i satisfies α i + β π is glued to v. If more than one verte is glued to v, then the folding would not be a zipping, as v would then constitute a junction of degree > in the gluing tree ([DO07, Sec..]). Note that if u i glues to v, then is determined: halfwa between u i and v along the perimeter of P. Thus we onl need tr each u i in turn, and check that Aleandrov s conditions hold for the uniquel determined zipping [DO07, Thm...]). This incidentall shows that an P with a refle verte admits onl O(n) zippings. For eample, appling this algorithm to the cube Z -unfolding in Figure results in si zippings: two copies of the one shown in that figure, two copies of a tetrahedron, one -verte and one -verte polhedron. Although this provides a linear-time algorithm for determining all zippings of P, it does not tell us which of these zippings lead to flat polhedra. Although
(c) Figure : The second Hamiltonian cut path on a cube. The resulting Hamiltonian S unfolding. (c) Zipped according to the indicated point identifications to a parallelogram polhedron of side lengths and.
Snapshots from an animation folding the parallelogram in Fig- Figure : ure (b,c).
(c) Figure : The third Hamiltonian cut path on a cube. The resulting Hamiltonian Z unfolding. (c) Zipped according to the indicated point identifications to a parallelogram polhedron of side lengths and. ½ 0 Figure 7: A zip-pair, abstractl. The perimeter has been normalized to. 7
there is an O(n ) algorithm for deciding if an Aleandrov gluing is flat [O R0], this remains unimplemented. We resorted to manual folding of the zippings.. Octahedron The octahedron also has three distinct Hamiltonian paths, one between the top and bottom vertices (separated b distance in the -skeleton), and two paths between adjacent (distance-) vertices. The first Hamiltonian unfolding both zips to a rectangle as shown in Figure 8, and zips to a parallelogram, Figure 9. I find the rectangle zipping especiall surprising, as it derives from a shape all of whose angles are multiples of π/ = 0. (c) Figure 8: Hamiltonian cut path on an octahedron. Its corresponding Hamiltonian unfolding. (c) Zipping folds it to a flat doubl covered rectangle of dimensions. One of the other Hamiltonian unfoldings of the octahedron, shown in Figure 0, zips to a parallelogram. The other Hamiltonian unfolding does not have This is natural because the cube and octahedron are duals. However, it is shown in [LDD + 0, Fig. ] that the dual of a Hamiltonian unfolding is not necessaril a Hamiltonian path through the faces of that unfolding. 8
Figure 9: Another zipping of the same unfolding from Figure 8 leads to a parallelogram polhedron. a flat zipping, although it does have zippings, e.g., to a tetrahedron all four of whose vertices have curvature π. I cannot resist mentioning that this last net folds to a flat rectangular polhedron, whose cut tree, however, is not a zipping: Figure.. Dodecahedron Ever Hamiltonian unfolding of the dodecahedron is zip-rigid, and therefore it has no flat zip pair in the sense posed in our Question above. The reason is as follows. Let and be the endpoints of the Hamiltonian path that unfolds the dodecahedron. Then the refle angle of the net at and is π =, leaving an eternal angle of there. The smallest conve angle in an edge unfolding of the dodecahedron is π = 08, so no verte can glue into or. Therefore, a zipping must zip at and, leading directl back to the dodecahedron. We should mention that loosening the criteria posed in our Question leads to a flat refolding of a Hamiltonian net for the dodecahedron. Figure illustrates one such, using the unfolding in Fig. in [LDD + 0]. Here the refolding in Figure (c) is neither conve nor a zipper folding. 9
(c) Figure 0: Hamiltonian cut path on an octahedron. Its corresponding Hamiltonian unfolding. (c) Zipping folds it to a flat doubl covered parallelogram of dimensions. 0
Figure : The same Hamiltonian unfolding from Figure 0 folds to a doubl covered rectangle, but this folding is not a zipping.
(c) a b Figure : Hamiltonian cut path on a dodecahedron. Its corresponding Hamiltonian unfolding. (c) A non-zipper refolding to a doubl covered flat nonconve polgon. The cut tree has degree at vertices a and b.. Icosahedron For the tetrahedron, cube, and octahedron, it was eas to eplore all the Hamiltonian unfoldings, because there are so few (,, and respectivel). The icosahedron, however, has hundreds of Hamiltonian unfoldings. At this writing, I do not know precisel how man geometricall distinct Hamiltonian unfoldings it possesses. The diameter of the icosahedral graph is, so the end points of a Hamiltonian path are a distance,, or apart. Fiing two vertices separated b a distance d {,, }, I found that there are, respectivel,, 08, and 70 labeled Hamiltonian paths between them. Of course not all these labeled paths are distinct geometric paths because of smmetries. However, I have not carried out the more difficult enumeration of the number of geometricall distinct (incongruent as paths in R ) Hamiltonian paths on an icosahedron. But certainl this number is less than + 08 + 70 = 80. For each of these 80 Hamiltonian unfoldings, I ran the zipping algorithm in Section., which determined that 8 of the unfoldings had at least one zipping, while all the others are zip-rigid (8 = + 0 + 0 in the three classes, respectivel). B visual inspection, it appears that of these Hamiltonian It is a curious fact that the number of labeled Hamiltonian ccles through an fied edge is 9 =. The simplicit of this epression suggests there might be a combinatorial eplanation, a question I asked on MathOverflow, http://mathoverflow.net/questions/7788/.
unfoldings are distinct; the are displaed in Figure. Figure : The distinct Hamiltonian unfolding of the icosahedron that each have at least one zipping to another conve polhedron (Not all are displaed to the same scale.) At least one of these unfoldings (the leftmost in the first row) zips to a parallelogram, as shown in Figure. At this writing we are uncertain if this is the onl zipping to a flat polhedron among the zipping unfoldings. Future Work As is evident from the foregoing, there is little theor behind the unfoldings detailed here. The central open problem is to gain more insight into which polhedra are net pairs, or more specificall, zipper pairs. Perhaps intuition can be strengthened b tackling specific subquestions that fall under this general umbrella. It is eas to list such questions, all of are open because of the lack of a general theor. For eample, the Hamiltonian unfoldings of the Archimedean solids detailed in [LDD + 0] could be eplored. An interesting specific but tangential question raised b this work is to determine the eact number of geometricall distinct Hamiltonian paths on a regular icosahedron. Acknowledgments. I thank Stephanie Annessi and Katherine Lipow for help in enumerating and folding the icosahedron Hamiltonian unfoldings. References [DDLO00] Erik D. Demaine, Martin L. Demaine, Anna Lubiw, and Joseph O Rourke. Eamples, countereamples, and enumeration re-
7 8 9 7 8 9 0 0 (c) 7 8 9 0 Figure : Hamiltonian cut path on an icosahedron. Its corresponding Hamiltonian unfolding. (c) Rezipping folds it to a flat doubl covered parallelogram of side lengths and.
sults for foldings and unfoldings between polgons and poltopes. Technical Report 09, Smith College, Northampton, Jul 000. arxiv:cs.cg/000709. [DDLO0] Erik D. Demaine, Martin L. Demaine, Anna Lubiw, and Joseph O Rourke. Enumerating foldings and unfoldings between polgons and poltopes. Graphs and Combin., 8():9 0, 00. See also [DDLO00]. [DO07] Erik D. Demaine and Joseph O Rourke. Geometric Folding Algorithms: Linkages, Origami, Polhedra. Cambridge Universit Press, Jul 007. http://www.gfalop.org. [LDD + 0] Anna Lubiw, Erik Demaine, Martin Demaine, Arlo Shallit, and Jonah Shallit. Zipper unfoldings of polhedral complees. In Proc. nd Canad. Conf. Comput. Geom., pages 9, August 00. [LO9] [O R0] Anna Lubiw and Joseph O Rourke. When can a polgon fold to a poltope? Technical Report 08, Dept. Comput. Sci., Smith College, June 99. Presented at Amer. Math. Soc. Conf., Oct. 99. Joseph O Rourke. On flat polhedra deriving from aleandrov s theorem. http://ariv.org/abs/007.0, Jul 00. [Pak0] Igor Pak. Lectures on discrete and polhedral geometr. http: //www.math.ucla.edu/~pak/book.htm, 00. [She7] Geoffre C. Shephard. Conve poltopes with conve nets. Math. Proc. Camb. Phil. Soc., 78:89 0, 97.