Universal Hinge Patterns for Folding Strips Efficiently into Any Grid Polyhedron

  • Nadia M. Benbernou
  • Erik D. DemaineEmail author
  • Martin L. Demaine
  • Anna Lubiw
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere. These geometric results offer a new way to build programmable matter that is substantially more efficient than what is possible with a square \(N \times N\) sheet of material, which can fold into all polycubes only of surface area O(N) and may stack \(\varTheta (N^2)\) layers at one point. We also show how our strip foldings can be executed by a rigid motion without collisions (albeit assuming zero thickness), which is not possible in general with 2D sheet folding.

To achieve these results, we develop new approximation algorithms for milling the surface of a grid polyhedron, which simultaneously give a 2-approximation in tour length and an 8 / 3-approximation in the number of turns. Both length and turns consume area when folding a strip, so we build on past approximation algorithms for these two objectives from 2D milling.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    An, B., Benbernou, N., Demaine, E.D., Rus, D.: Planning to fold multiple objects from a single self-folding sheet. Robotica 29(1), 87–102 (2011). Special issue on Robotic Self-X SystemsGoogle Scholar
  2. 2.
    Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal covering tours with turn costs. SIAM Journal on Computing 35(3), 531–566 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arkin, E.M., Fekete, S.P., Mitchell, J.S.B.: Approximation algorithms for lawn mowing and milling. Computational Geometry: Theory and Applications 17(1–2), 25–50 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benbernou, N.M.: Geometric Algorithms for Reconfigurable Structures. PhD thesis. Massachusetts Institute of Technology, September 2011Google Scholar
  5. 5.
    Benbernou, N.M., Demaine, E.D., Demaine, M.L., Lubiw, A.: Universal hinge patterns for folding strips efficiently into any grid polyhedron (2016).
  6. 6.
    Benbernou, N.M., Demaine, E.D., Demaine, M.L., Ovadya, A.: Universal hinge patterns to fold orthogonal shapes. In: Origami\(^5\): Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education, pp. 405–420. A K Peters, Singapore (2010)Google Scholar
  7. 7.
    Cheung, K.C., Demaine, E.D., Bachrach, J., Griffith, S.: Programmable assembly with universally foldable strings (moteins). IEEE Transactions on Robotics 27(4), 718–729 (2011)CrossRefGoogle Scholar
  8. 8.
    Clementi, A.E.F., Crescenzi, P., Rossi, G.: On the complexity of approximating colored-graph problems extended abstract. In: Asano, T., Imai, H., Lee, D.T., Nakano, S., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 281–290. Springer, Heidelberg (1999). doi: 10.1007/3-540-48686-0_28 CrossRefGoogle Scholar
  9. 9.
    Demaine, E.D., Demaine, M.L., Mitchell, J.S.B.: Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. Computational Geometry: Theory and Applications 16(1), 3–21 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demaine, E.D., Tachi, T.: Origamizer: A practical algorithm for folding any polyhedron (2017) (manuscript)Google Scholar
  11. 11.
    Genc, B.: Reconstruction of Orthogonal Polyhedra. PhD thesis. University of Waterloo (2008)Google Scholar
  12. 12.
    Hawkes, E., An, B., Benbernou, N.M., Tanaka, H., Kim, S., Demaine, E.D., Rus, D., Wood, R.J.: Programmable matter by folding. Proceedings of the National Academy of Sciences of the United States of America 107(28), 12441–12445 (2010)CrossRefGoogle Scholar
  13. 13.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6(3), 243–254 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lang, R. J.: A computational algorithm for origami design. In: Proceedings of the 12th Annual ACM Symposium on Computational Geometry, Philadelphia, PA, pp. 98–105, May 1996Google Scholar
  15. 15.
    Lang, R.J., Demaine, E.D.: Facet ordering and crease assignment in uniaxial bases. In: Origami\(^4\): Proceedings of the 4th International Conference on Origami in Science, Mathematics, and Education, Pasadena, California, pp. 189–205. A K Peters, September 2006Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Nadia M. Benbernou
    • 1
  • Erik D. Demaine
    • 2
    Email author
  • Martin L. Demaine
    • 2
  • Anna Lubiw
    • 3
  1. 1.Google Inc.Menlo ParkUSA
  2. 2.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  3. 3.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations