Algebraic Analysis of Huzita’s Origami Operations and Their Extensions

  • Fadoua Ghourabi
  • Asem Kasem
  • Cezary Kaliszyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7993)


We investigate the basic fold operations, often referred to as Huzita’s axioms, which represent the standard seven operations used commonly in computational origami. We reformulate the operations by giving them precise conditions that eliminate the degenerate and incident cases. We prove that the reformulated ones yield a finite number of fold lines. Furthermore, we show how the incident cases reduce certain operations to simpler ones. We present an alternative single operation based on one of the operations without side conditions. We show how each of the reformulated operations can be realized by the alternative one. It is known that cubic equations can be solved using origami folding. We study the extension of origami by introducing fold operations that involve conic sections. We show that the new extended set of fold operations generates polynomial equations of degree up to six.


fold operations computational origami conic section 


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  1. 1.
    Alperin, R.C.: A Mathematical Theory of Origami Constructions and Numbers. New York Journal of Mathematics 6, 119–133 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alperin, R.C., Lang, R.J.: One-, Two, and Multi-fold Origami Axioms. In: Origami4 Fourth International Meeting of Origami Science, Mathematics and Education (4OSME), pp. 371–393. A K Peters Ltd. (2009)Google Scholar
  3. 3.
    Ghourabi, F., Ida, T., Takahashi, H., Kasem, A.: Reasoning Tool for Mathematical Origami Construction. In: CD Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC 2009 (2009)Google Scholar
  4. 4.
    Huzita, H.: Axiomatic Development of Origami Geometry. In: Proceedings of the First International Meeting of Origami Science and Technology, pp. 143–158 (1989)Google Scholar
  5. 5.
    Ida, T., Kasem, A., Ghourabi, F., Takahashi, H.: Morley’s theorem revisited: Origami construction and automated proof. Journal of Symbolic Computation 46(5), 571–583 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jones, A., Morris, S.A., Pearson, K.R.: Abstract Algebra and Famous Impossibilities. Springer-Verlag New York, Inc. (1991)Google Scholar
  7. 7.
    Justin, J.: Résolution par le pliage de l’équation du troisième degré et applications géométriques. In: Proceedings of the First International Meeting of Origami Science and Technology, pp. 251–261 (1989)Google Scholar
  8. 8.
    Kaliszyk, C., Ida, T.: Proof Assistant Decision Procedures for Formalizing Origami. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/MKM 2011. LNCS, vol. 6824, pp. 45–57. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    Kasem, A., Ghourabi, F., Ida, T.: Origami Axioms and Circle Extension. In: Proceedings of the 26th Symposium on Applied Computing (SAC 2011), pp. 1106–1111. ACM Press (2011)Google Scholar
  10. 10.
    Martin, G.E.: Geometric Constructions. Springer-Verlag New York, Inc. (1998)Google Scholar
  11. 11.
    Wantzel, P.L.: Recherches sur les moyens de connaître si un problème de géométrie peut se résoudre avec la règle et le compas. Journal de Mathématiques Pures et Appliquées, 366–372 (1984)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fadoua Ghourabi
    • 1
  • Asem Kasem
    • 2
  • Cezary Kaliszyk
    • 3
  1. 1.Kwansei Gakuin UniversityJapan
  2. 2.Taylor’s UniversityMalaysia
  3. 3.University of InnsbruckAustria

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