Pseudo-Edge Unfoldings of Convex Polyhedra

  • Nicholas Barvinok
  • Mohammad GhomiEmail author
Ricky Pollack Memorial Issue


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.


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 



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.


  1. 1.
    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  2. 2.
    Barvinok, N., Ghomi, M.: Pseudo-edge.nb (2017). Mathematica Package.
  3. 3.
    Bern, M., Demaine, E.D., Eppstein, D., Kuo, E., Mantler, A., Snoeyink, J.: Ununfoldable polyhedra with convex faces. Comput. Geom. 24(2), 51–62 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bobenko, A.I., Izmestiev, I.: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447–505 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y.: Edge-Unfolding Almost-Flat Convex Polyhedral Terrains. Master’s Thesis (2013)Google Scholar
  6. 6.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Problem Books in Mathematics. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations. Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)zbMATHGoogle Scholar
  8. 8.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dürer, A.: The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Drer for the Use of All Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV. Abaris Books, New York (1977) (1525)Google Scholar
  10. 10.
    Ghomi, M.: A Riemannian four vertex theorem for surfaces with boundary. Proc. Am. Math. Soc. 139(1), 293–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ghomi, M.: Affine unfoldings of convex polyhedra. Geom. Topol. 18(5), 3055–3090 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ghomi, M.: Dürer’s unfolding problem for convex polyhedra. Not. Am. Math. Soc. 65(1), 25–27 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grünbaum, B.: Nets of polyhedra. II. Geombinatorics 1(3), 5–10 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Izmestiev, I.: Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry (2017). arXiv:1707.02172
  15. 15.
    Lubiw, A., O’Rourke, J.: Angle-monotone paths in non-obtuse triangulations (2017). arXiv:1707.00219
  16. 16.
    O’Rourke, J.: How to Fold It: The Mathematics of Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    O’Rourke, J.: Unfolding convex polyhedra via radially monotone cut trees (2016). arXiv:1607.07421
  18. 18.
    O’Rourke, J.: Edge-unfolding nearly flat convex caps (2017). arXiv:1707.01006
  19. 19.
    O’Rourke, J.: Addendum to: Edge-unfolding nearly flat convex caps (2017). arXiv:1709.02433
  20. 20.
    Pak, I.: Lectures on Discrete and Polyhedral Geometry (2008)Google Scholar
  21. 21.
    Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. Translations of Mathematical Monographs, vol. 35. American Mathematical Society, Providence (1973)zbMATHGoogle Scholar
  22. 22.
    Schulz, A.: Lifting Planar Graphs to Realize Integral 3-Polytopes and Topics in Pseudo-Triangulations. Ph.D. Thesis (2008)Google Scholar
  23. 23.
    Shephard, G.C.: Convex polytopes with convex nets. Math. Proc. Camb. Philos. Soc. 78(3), 389–403 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tarasov, A.: Existence of a polyhedron which does not have a non-overlapping pseudo-edge unfolding (2008). arXiv:0806.2360v3
  25. 25.
    Whiteley, W.: Motions and stresses of projected polyhedra. Struct. Topol. 1982(7), 13–38 (1982)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations