Skip to main content
SpringerLink
Log in

Geodesic Universal Molecules

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

The first phase of TreeMaker, a well-known method for origami design, decomposes a planar polygon (the “paper”) into regions. If some region is not convex, TreeMaker indicates it with an error message and stops. Otherwise, a second phases is invoked which computes a crease pattern called a “universal molecule”. In this paper we introduce and study geodesic universal molecules, which also work with non-convex polygons and thus extend the applicability of the TreeMaker method. We characterize the family of disk-like surfaces, crease patterns and folded states produced by our generalized algorithm. They include non-convex polygons drawn on the surface of an intrinsically flat piecewise-linear surface which have self-overlap when laid open flat, as well as surfaces with negative curvature at a boundary vertex.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aichholzer O., Alberts D., Aurenhammer F., Gärtner B.: A novel type of skeleton for polygons. J. Univers. Comput. Sci. 1(12), 752–761 (1995)

    MathSciNet  MATH  Google Scholar 

  2. Aloupis G., Demaine E.D., Langerman S., Morin P., O’Rourke J., Streinu I., Toussaint G.T.: Edge-unfolding nested polyhedral bands. Comput. Geom. Theory Appl. 39(1), 30–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alperin R., Hayes B., Lang R.: Folding the hyperbolic crane. Math. Intell. 34(2), 38–49 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aurenhammer F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. (CSUR) 23(3), 345–405 (1991)

    Article  Google Scholar 

  5. Bowers, J., Streinu, I.: Lang’s universal molecule algorithm. Ann. Math. Artif. Intell. 1–30 (2014)

  6. Bowers, J.C., Streinu, I.: Rigidity of origami universal molecules. In: Ida, T., Fleuriot, J.D. (eds.) Automated Deduction in Geometry. Lecture Notes in Computer Science, vol. 7993, pp. 120–142. Springer, New York (2012)

  7. Bowers, J.C., Streinu, I.: Computing origami universal molecules with cyclic tournament forests. In: Proc. 15th Intern. Symp. on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC’13), pp. 42–52. IEEE (2013)

  8. Demaine, E.D., Demaine, M.L.: Computing extreme origami bases. Technical Report CS-97-22, Dept. of Computer Science, University of Waterloo, Waterloo (1997)

  9. Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle packing for origami design is hard. In: Wang-Iverson, P., Lang, R.J., Yim, M. (eds.) Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education, pp. 609–626. Taylor and Francis, London (2011)

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

    Book  MATH  Google Scholar 

  11. Ida T., Ghourabi F., Takahashi K.: Formalizing polygonal knot origami. J. Symb. Comput. 69, 93–108 (2015)

    Article  MathSciNet  MATH  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)

  13. Lang, R.J.: Treemaker 4.0: A program for origami design. http://www.langorigami.com (1998)

  14. Tachi T.: Origamizing polyhedral surfaces. IEEE Trans. Vis. Comput. Graph. 16(2), 298–311 (2010)

    Article  Google Scholar 

  15. Tanaka H.: Bi-stiffness property of motion structures transformed into square cells. Proc. R. Soc. A Math. Phys. Eng. Sci. 469(2156), 20130063 (2013)

    Article  Google Scholar 

  16. Wu W., You Z.: Modelling rigid origami with quaternions and dual quaternions. Proc. R. Soc. A Math. Phys. Eng. Sci. 466(2119), 2155–2174 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu, W., You, Z.: A solution for folding rigid tall shopping bags. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 467, pp. 2561–2574. The Royal Society (2011)

  18. Yasuda H., Yein T., Tachi T., Miura K., Taya M.: Folding behaviour of Tachi–Miura polyhedron bellows. Proc. R. Soc. A Math. Phys. Eng. Sci. 469(2159), 20130351 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, James Madison University, Harrisonburg, VA, 22801, USA

    John C. Bowers

  2. Department of Computer Science, Smith College, Northampton, MA, 01063, USA

    Ileana Streinu

Authors
  1. John C. Bowers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ileana Streinu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ileana Streinu.

Additional information

We acknowledge support for this work through an NSF graduate fellowship (JB) and NSF Grant CCF-1319366 (IS).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bowers, J.C., Streinu, I. Geodesic Universal Molecules. Math.Comput.Sci. 10, 115–141 (2016). https://doi.org/10.1007/s11786-016-0253-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-016-0253-5

Keywords

Mathematics Subject Classification

Search

Navigation