Mathematics in Computer Science

, Volume 10, Issue 1, pp 115–141 | Cite as

Geodesic Universal Molecules

  • John C. Bowers
  • Ileana StreinuEmail author


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.


Algorithmic origami Planar subdivision Metric tree Non-convex polygon 

Mathematics Subject Classification

68U05 (Computer graphics; computational geometry) 


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  1. 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)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alperin R., Hayes B., Lang R.: Folding the hyperbolic crane. Math. Intell. 34(2), 38–49 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aurenhammer F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. (CSUR) 23(3), 345–405 (1991)CrossRefGoogle Scholar
  5. 5.
    Bowers, J., Streinu, I.: Lang’s universal molecule algorithm. Ann. Math. Artif. Intell. 1–30 (2014)Google Scholar
  6. 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)Google Scholar
  7. 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)Google Scholar
  8. 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)Google Scholar
  9. 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)Google Scholar
  10. 10.
    Demaine E.D., O’Rourke J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ida T., Ghourabi F., Takahashi K.: Formalizing polygonal knot origami. J. Symb. Comput. 69, 93–108 (2015)MathSciNetCrossRefzbMATHGoogle 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)
  14. 14.
    Tachi T.: Origamizing polyhedral surfaces. IEEE Trans. Vis. Comput. Graph. 16(2), 298–311 (2010)CrossRefGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)Google Scholar
  18. 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)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceJames Madison UniversityHarrisonburgUSA
  2. 2.Department of Computer ScienceSmith CollegeNorthamptonUSA

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