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Pseudo-Edge Unfoldings of Convex Polyhedra

  • Ricky Pollack Memorial Issue
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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.

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Acknowledgements

Our debt to the original investigations of A. Tarasov [24] is evident throughout this work. Thanks also to J. O’Rourke for his interest and useful comments on earlier drafts of this paper. Furthermore we are grateful to several anonymous reviewers who prompted us to clarify the exposition of this work. Parts of this work were completed while the first named author participated in the REU program in the School of Math at Georgia Tech in the Summer of 2017.

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  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Nicholas Barvinok & Mohammad Ghomi

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  1. Nicholas Barvinok
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  2. Mohammad Ghomi
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Correspondence to Mohammad Ghomi.

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Dedicated to the memory of Ricky Pollack.

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Research of the second named author was supported in part by NSF Grant DMS–1308777.

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Barvinok, N., Ghomi, M. Pseudo-Edge Unfoldings of Convex Polyhedra. Discrete Comput Geom 64, 671–689 (2020). https://doi.org/10.1007/s00454-019-00082-1

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