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

O'Rourke, Joseph

English

arXiv

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

O\u27Rourke, Joseph

Smith College: Smith ScholarWorks

Masthead Logo Smith ScholarWorksComputer Science: Faculty Publications Computer Science7-19-2010On Folding a Polygon to a PolyhedronJoseph O'RourkeSmith College, jorourke@smith.eduFollow this and additional works at: https://scholarworks.smith.edu/csc_facpubsPart of the Computer Sciences Commons, and the Geometry and Topology CommonsThis Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. Formore information, please contact scholarworks@smith.eduRecommended CitationO'Rourke, Joseph, "On Folding a Polygon to a Polyhedron" (2010). Computer Science: Faculty Publications, Smith College,Northampton, MA.https://scholarworks.smith.edu/csc_facpubs/35On Folding a Polygon to a PolyhedronJoseph O’Rourke∗July 20, 2010AbstractWe show that the open problem presented in Geometric Folding Algo-rithms: Linkages, Origami, Polyhedra [DO07] is solved by a theorem ofBurago and Zalgaller [BZ96] from more than a decade earlier.1 IntroductionIn [DO07, p. 384] we formulated this open problem:Open Problem 25.1: Folding Polygons to (Nonconvex) PolyhedraDoes every simple polygon fold (by perimeter gluing) to some simple poly-hedron? In particular, does some perimeter halving always lead to a poly-hedron?This note explains how this problem is solved (positively) by a theoremin [BZ96]. There is nothing original in this note. It is merely a description ofearlier work, interpreting it in the context of the folding problem.The theorem of Burago and Zalgaller [BZ96, p. 370] that solves the problemis this:Theorem 1 (Burago-Zalgaller 1.7) “Every polyhedron M admits an isomet-ric piecewise-linear C0 immersion into R3. If M is orientable or has a nonemptyboundary, then M admits an isometric piecewise-linear C0 embedding into R3.”Interpreting this theorem (henceforth, the BZ theorem) and connecting it tothe folding problem requires an analysis of its technical terms.∗Department of Computer Science, Smith College, Northampton, MA 01063, USA.orourke@cs.smith.edu.1arXiv:1007.3181v1  [cs.CG]  19 Jul 20102 Definition of TermsFirst we explain the folding problem from [DO07]. Folding a polygon by perime-ter gluing means identifying points of the boundary of the polygon with otherpoints so that the resulting manifold is topologically a sphere, that is, closed,orientable, and with genus zero. Isolated points might have no match, which oc-curs where the boundary is “zipped” in the neighborhood of such a point. Thisis perhaps clearest for a perimeter halving. Let x and y be points on the bound-ary of the polygon that partition the perimeter into two equal-length halves.Then perimeter halving identifies the two halves of the boundary, zipping in theneighborhoods of x and y. (See ahead to Figure 1.) Our question was whetherevery perimeter gluing, and more specifically, every perimeter halving, produces(is realized by) a simple (non-self-intersecting) polyhedron.This question is an analog of Alexandrov’s 1941 theorem, described in [DO07,Sec. 23.3], [Pak10, Sec. 37], and Alexandrov’s book [Ale05]. Alexandrov’s theo-rem adds one more condition—that the gluing creates no more than 2π surfaceangle surrounding any point of the resulting manifold—and concludes that theresult is a unique convex polyhedron, where “polyhedron” is interpreted to in-clude a flat doubly covered convex polygon. The focus of our problem was toask for an extension without the 2π convexity condition. The restriction tofolding a single polygon is not essential to the problem. Alexandrov considereda more general gluing of a collection of polygons, which has been translated asa gluing of a “development” [Ale05, p. 50ff], terminology not always followed bylater authors.We will draw several definitions from [Pak06] (many also incorporatedinto [Pak10]), for they seem the clearest on this topic. In particular, the notionof a “gluing” is captured in this definition [Pak06, Sec. 1.3]:“Let S be an abstract 2-dimensional polyhedral surface defined asa collection of triangles T1, . . . , Tm with combinatorial gluing rules.Here each triangle Ti is given by its edge lengths, and whenevertwo edges are glued, they have equal length. We always assumethat S is a connected simplicial complex, and that it is closed (hasno boundary) and orientable, i.e. homeomorphic to a sphere withg ≥ 0 handles.”Restricting to g = 0 yields exactly the gluings that are our main focus. Ourpolygon can be partitioned into triangles so that the perimeter gluing is capturedby edge-to-edge gluings of the triangles. To simplify the language, we will callPak’s “abstract 2-dimensional polyhedral surface” a gluing of polygons. Nextwe concentrate on the phrase “every polyhedron M” in the BZ theorem, andshow that it refers to gluings of polygons. From [BZ96, p. 369]:“By a two-dimensional manifold with polyhedral metric (in brief,a polyhedron), we mean a metric space endowed with the structureof a connected compact two-dimensional manifold (possibly withboundary) every point x of which has a neighborhood isometric to2the vertex of a cone. ... The metric is locally flat everywhere exceptfor a finite collection of points; these points are the ‘true’ vertices.”Although their naming this concept a “polyhedron” is nonstandard, it is clearthis coincides with the notion of a gluing of polygons. The “true vertices” Vof such a gluing are the points whose total angle differs from 2π, at which thediscrete curvature is concentrated.For definitions of “immersion” and “embedding,” we again turn toPak [Pak06]:“A (3-dimensional) realization of S is defined as a map f : V 7→ R3such that the Euclidean distance ||v1, v2|| between vertices is equalto the edge length |v1, v2| of any triangle Ti which contains v1 andv2.An immersion is a realization where no two triangles have a 2-dimensional intersection. For example, a doubly covered triangle isa realization in R3 of a surface homeomorphic to a sphere, but notan immersion. An embedding is a realization where two trianglesintersect only by an edge or by a vertex they share. We always con-sider surfaces S up to isometry, so we speak of isometric immersionsand isometric embeddings.”Our interest in the folding problem is focused on embeddings (with the ex-ception of doubly covered polygons): because we imagine performing the foldingwith paper, we do not want the paper to penetrate itself. So we henceforth con-centrate on the second half of the BZ theorem, which describes embeddings.In our context, that the embedding is C0 simply means that it is continu-ous, that is, without tearing. I believe they include this qualification primarilyto distinguish their result from the famous C1-smooth embedding theorem ofNash [Nas54] (improved by Kuiper [Kui55]). This Nash-Kuiper result cannotbe improved to C2. Histories of immersion and embedding theorems can befound in [Spr05] and [Sau10].The key phrase in the BZ theorem is that the embedding is “isometricpiecewise-linear.” Isometric simply means distances within the surface are main-tained. Piecewise-linear means that each triangle Ti from the gluing of polygonsis mapped to a finite collection of triangles in R3. Continuous isometric defor-mations are often called bendings in the literature (e.g., [Pak10, Sec. 38.1]). Theaddition of “piecewise-linear” means that the bending is accomplished with flatpieces. A nice phrase used in [BZ96, Lem. 2.2] to describe the mapping of onetriangle is that it becomes a “‘pleated’ surface” in R3. The triangle gets creasedalong a network of straight segments. From the construction in that paper, itis clear that only a finite number of pieces are used to embed each triangle Ti.So we may rephrase the embedding part of the BZ theorem that is our focusas follows:Theorem 2 Every gluing of polygons to form an oriented manifold has an iso-metric embedding as a polyhedral surface (composed of a finite number of flattriangles) in R3.3Because perimeter gluing, and perimeter halving in particular, constitutesa gluing of polygons, this theorem provides a positive answer to Open Prob-lem 25.1.I presented Figure 1 at the 20th Symposium on Computational Geometry(the proceedings cover image) as a challenge to fold to a polyhedron. The figuremarks out a particular perimeter-halving folding, which, by BZ’s theorem, mustfold to a simple polyhedron.yxFigure 1: Points x and y are perimeter-halving points. The folding “zips” theperimeter closed from x to y. A short initial segment of the gluing in theneighborhood of x is indicated: the red portion of the boundary glues to thesymmetric blue portion. This perimeter gluing continues from x to y.3 DiscussionOne of the motivations of the work in [BZ96] was to show that one can increasethe volume of a convex polyhedron by making it nonconvex, but otherwise re-maining isometric to the original. This question is explored in great depthin [Pak06] and [Pak08]. The constructions are piecewise-linear, that is, polyhe-dral.Although I have been citing [BZ96] as resolving the folding problem, in factearlier (as yet untranslated) work of Burago and Zalgaller [BZ60] apparently4already sufficed for the special case when the manifold is homeomorphic to asphere. This is cited in support of Exercise 39.13(a) in [Pak10].Although the problem is solved by the BZ theorem, their proof is sufficientlycomplex that it is difficult to see what their construction will produce for aspecific example, such as the folding in Figure 1. Saucan has a useful summaryof the Burago-Zalgaller construction in [Sau10]. The construction incorporatesideas from Nash’s C1 embedding construction. In particular, Nash’s “‘spiralling’perturbations” [Nas54, p. 383] of the surface results in a polyhedral surface thatis “strongly ‘corrugated’,” to quote Saucan’s apt description. Although thenumber of triangular facets of the final embedded polyhedron is finite, it doesnot seem straightforward to provide an explicit upper bound on the number offacets. The work of Bern and Hayes in [BH09] achieves a flat embedding in R2with just O(n) facets, where n is the number of triangles of the polygon gluing.Perhaps their work will lead to a more efficient construction in R3.Acknowledgments. I thank Igor Pak for connecting me to the relevant litera-ture, and patiently explaining key definitions. Any remaining misinterpretationsare solely my own.References[Ale05] Aleksandr D. Alexandrov. Convex Polyhedra. Springer-Verlag, Berlin,2005. Monographs in Mathematics. Translation of the 1950 Russianedition by N. S. Dairbekov, S. S. Kutateladze, and A. B. Sossinsky.[BH09] Marshall Bern and Barry Hayes. Origami embedding of piecewise-lineartwo-manifolds. In Proc. 8th Internat. Latin Amer. Symp. TheoreticalInformatics, pages 617–629. Springer LNCS 4957, 2009.[BZ60] Ju. D. Burago and V. A. Zalgaller. Polyhedral embedding of a net.Vestnik Leningrad. Univ, 15(7):66–80, 1960. In Russian.[BZ96] Yu. D. Burago and V. A. Zalgaller. Isometric piecewise linear immer-sions of two-dimensional manifolds with polyhedral metrics into R3. St.Petersburg Math. J., 7(3):369–385, 1996. Translated by S. G. Ivanov.[DO07] Erik D. Demaine and Joseph O’Rourke. Geometric Folding Algorithms:Linkages, Origami, Polyhedra. Cambridge University Press, July 2007.http://www.gfalop.org.[Kui55] Nicolaas H. Kuiper. On C1-isometric imbeddings: I, II. Indag. Math.,17:545–556, 683–689, 1955. Also Nederl. Akad. Wetensch. Proc. Ser.A., 58.[Nas54] John Nash. C1 isometric imbeddings. Ann. Math., 60(3):383–396, 1954.[Pak06] Igor Pak. Inflating polyhedral surfaces. http://www.math.ucla.edu/~pak/papers/pillow4.pdf, 2006.5[Pak08] Igor Pak. Inflating the cube without stretching. Amer. Math. Monthly,115(5):443–445, 2008.[Pak10] Igor Pak. Lectures on Discrete and Polyhedral Geometry. http://www.math.ucla.edu/~pak/book.htm, 2010.[Sau10] Emil Saucan. Isometric embeddings in imaging and vision: Facts andfiction. arXiv:1004.5351v2 [cs.CV], 2010.[Spr05] David Spring. The golden age of immersion theory in topology: 1959-1973. A mathematical survey from a historical perspective. Bull. Amer.Math. Soc., 42:163–180, 2005.6

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