Rigidity of Origami Universal Molecules

  • John C. Bowers
  • Ileana Streinu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7993)


In a seminal paper from 1996 that marks the beginning of computational origami, R. Lang introduced TreeMaker, a method for designing origami crease patterns with an underlying metric tree structure. In this paper we address the foldability of paneled origamis produced by Lang’s Universal Molecule algorithm, a key component of TreeMaker.

We identify a combinatorial condition guaranteeing rigidity, resp. stability of the two extremal states relevant to Lang’s method: the initial flat, open state, resp. the folded origami base computed by Lang’s algorithm. The proofs are based on a new technique of transporting rigidity and flexibility along the edges of a paneled surface.


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  1. 1.
    Bern, M.W., Hayes, B.: The complexity of flat origami. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (1996)Google Scholar
  2. 2.
    Bern, M.W., Hayes, B.: Origami embedding of piecewise-linear two-manifolds. Algorithmica 59(1), 3–15 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bowers, J.C., Streinu, I.: Lang’s universal molecule algorithm (video). In: Proc. 28th Symp. Computational Geometry, SoCG 2012 (2012)Google Scholar
  4. 4.
    Bowers, J.C., Streinu, I.: Lang’s universal molecule algorithm. Technical report, University of Massachusetts and Smith College (December 2011)Google Scholar
  5. 5.
    Bricard, R.: Mémoire sur la théorie de l’octaèdre articulé. J. Math. Pure et Appl. 5, 113–148 (1897)Google Scholar
  6. 6.
    Bricard, R.: Memoir on the theory of the articulated octahedron. Translation from the French original of [5] (March 2012)Google Scholar
  7. 7.
    Demaine, E.D., Demaine, M.L.: Computing extreme origami bases. Technical Report CS-97-22, Department of Computer Science, University of Waterloo (May 1997)Google Scholar
  8. 8.
    Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and one straight cut suffices. In: Proc. 10th Annual ACM-SIAM Sympos. Discrete Alg. (SODA 1999), pp. 891–892 (January 1999)Google Scholar
  9. 9.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press (2007)Google Scholar
  10. 10.
    Eppstein, D.: Faster circle packing with application to nonobtuse triangulations. International Journal of Computational Geometry and Applications 7(5), 485–491 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Farber, M.: Invitation to Topological Robotics. Zürich Lectures in Advanced Mathematics. European Mathematical Society (2008)Google Scholar
  12. 12.
    Lang, R.J.: A computational algorithm for origami design. In: Proceedings of the 12th Annual ACM Symposium on Computational Geometry, pp. 98–105 (1996)Google Scholar
  13. 13.
    Lang, R.J.: Treemaker 4.0: A program for origami design (1998)Google Scholar
  14. 14.
    Lang, R.J.: Origami design secrets: mathematical methods for an ancient art. A.K. Peters Series. A.K. Peters (2003)Google Scholar
  15. 15.
    Lang, R.J. (ed.): Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education. A.K. Peters (2009)Google Scholar
  16. 16.
    Panina, G., Streinu, I.: Flattening single-vertex origami: the non-expansive case. Computational Geometry: Theory and Applications 46(8), 678–687 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Streinu, I., Whiteley, W.: Single-vertex origami and spherical expansive motions. In: Akiyama, J., Kano, M., Tan, X. (eds.) JCDCG 2004. LNCS, vol. 3742, pp. 161–173. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Wang-Iverson, P., Lang, R.J., Yim, M.: Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education. Taylor and Francis (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • John C. Bowers
    • 1
  • Ileana Streinu
    • 2
  1. 1.Department of Computer ScienceUniversity of MassachusettsAmherstUSA
  2. 2.Department of Computer ScienceSmith CollegeNorthamptonUSA

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