Pseudo-Edge Unfoldings of Convex Polyhedra
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Pseudo-Edge Unfoldings of Convex Polyhedra Nicholas Barvinok1 · Mohammad Ghomi1 Received: 18 January 2018 / Revised: 4 February 2019 / Accepted: 13 March 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K , whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Dürer’s conjecture does not hold for pseudo-edge unfoldings. Keywords Edge unfolding · Dürer conjecture · Almost flat convex cap · Prescribed curvature · Weighted spanning forest · Pseudo-edge graph · Isometric embedding Mathematics Subject Classification Primary: 52B10 · 53C45 · Secondary: 57N35 · 05C10
1 Introduction By a convex polyhedron in this work we mean the boundary of the convex hull of finitely many points in Euclidean space R3 which do not all lie in a plane. A well-
Dedicated to the memory of Ricky Pollack. Editor in Charge: János Pach Research of the second named author was supported in part by NSF Grant DMS–1308777. Nicholas Barvinok [email protected] http://www.math.gatech.edu/∼nbarvinok3/ Mohammad Ghomi [email protected] http://www.math.gatech.edu/∼ghomi 1
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
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Discrete & Computational Geometry
Fig. 1
known conjecture [8], attributed to the Renaissance painter Albrecht Dürer [9], states that every convex polyhedron K is unfoldable, i.e., it may be cut along some spanning tree of its edges and isometrically embedded into the plane R2 . Here we study a generalization of this problem to pseudo-edges of K , i.e., distance minimizing geodesic segments in K connecting pairs of its vertices (see Fig. 1 for an example of a pseudoedge which is not an actual edge). A pseudo-edge graph E of K is a 3-connected embedded graph composed of pseudo-edges of K , with the same vertices as those of K , and with faces which are convex in K , i.e., the interior angles of each face of E are less than π . Cutting K along any spanning tree T of E yields a simply connected compact surface K T which admits an isometric immersion or unfolding u T : K T → R2 . If u T is one-to-one for some T , then we say that K is unfoldable with respect to E. The main result of this paper is as follows: Theorem 1.1 There exists a convex polyhedron K with 340 vertices and a pseudo-edge graph with respect to which K is not unfoldable. Thus one may say that Dürer’s conjecture does not hold in a purely intrinsic sense, since it is not possible to distinguish a pseudo-edge from an actual edge by means of local measurements within
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