Skip to main content

Rigidity of Origami Universal Molecules

  • Conference paper
Automated Deduction in Geometry (ADG 2012)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7993))

Included in the following conference series:

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bern, M.W., Hayes, B.: The complexity of flat origami. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (1996)

    Google Scholar 

  2. Bern, M.W., Hayes, B.: Origami embedding of piecewise-linear two-manifolds. Algorithmica 59(1), 3–15 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bowers, J.C., Streinu, I.: Lang’s universal molecule algorithm (video). In: Proc. 28th Symp. Computational Geometry, SoCG 2012 (2012)

    Google Scholar 

  4. Bowers, J.C., Streinu, I.: Lang’s universal molecule algorithm. Technical report, University of Massachusetts and Smith College (December 2011)

    Google Scholar 

  5. Bricard, R.: Mémoire sur la théorie de l’octaèdre articulé. J. Math. Pure et Appl. 5, 113–148 (1897)

    Google Scholar 

  6. Bricard, R.: Memoir on the theory of the articulated octahedron. Translation from the French original of [5] (March 2012)

    Google Scholar 

  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. 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. Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press (2007)

    Google Scholar 

  10. Eppstein, D.: Faster circle packing with application to nonobtuse triangulations. International Journal of Computational Geometry and Applications 7(5), 485–491 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farber, M.: Invitation to Topological Robotics. Zürich Lectures in Advanced Mathematics. European Mathematical Society (2008)

    Google Scholar 

  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. Lang, R.J.: Treemaker 4.0: A program for origami design (1998)

    Google Scholar 

  14. Lang, R.J.: Origami design secrets: mathematical methods for an ancient art. A.K. Peters Series. A.K. Peters (2003)

    Google Scholar 

  15. Lang, R.J. (ed.): Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education. A.K. Peters (2009)

    Google Scholar 

  16. Panina, G., Streinu, I.: Flattening single-vertex origami: the non-expansive case. Computational Geometry: Theory and Applications 46(8), 678–687 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bowers, J.C., Streinu, I. (2013). Rigidity of Origami Universal Molecules. In: Ida, T., Fleuriot, J. (eds) Automated Deduction in Geometry. ADG 2012. Lecture Notes in Computer Science(), vol 7993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40672-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40672-0_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40671-3

  • Online ISBN: 978-3-642-40672-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics