Advertisement

Computational Construction of a Maximum Equilateral Triangle Inscribed in an Origami

  • Tetsuo Ida
  • Hidekazu Takahashi
  • Mircea Marin
  • Fadoua Ghourabi
  • Asem Kasem
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)

Abstract

We present an origami construction of a maximum equilateral triangle inscribed in an origami, and an automated proof of the correctness of the construction. The construction and the correctness proof are achieved by a computational origami system called Eos (E-origami system). In the construction we apply the techniques of geometrical constraint solving, and in the automated proof we apply Gröbner bases theory and the cylindrical algebraic decomposition method. The cylindrical algebraic decomposition is indispensable to the automated proof of the maximality since the specification of this property involves the notion of inequalities. The interplay of construction and proof by Gröbner bases method and the cylindrical algebraic decomposition supported by Eos is the feature of our work.

Keywords

Geometrical Property Equilateral Triangle Geometric Constraint Correctness Proof Algebraic Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: The basic algorithm. SIAM Journal on Computing 13(4), 865–877 (1984)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Buchberger, B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes mathematicae 4(3), 374–383 (1970)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Buchberger, B., Dupre, C., Jebelean, T., Kriftner, F., Nakagawa, K., Văsaru, D., Windsteiger, W.: The Theorema Project: A Progress Report. In: Kerber, M., Kohlhase, M. (eds.) Symbolic Computation and Automated Reasoning: The Calculemus-2000 Symposium, St. Andrews, Scotland, August 6-7, pp. 98–113 (2000)Google Scholar
  4. 4.
    Demaine, E.D., Demaine, M.L.: Recent results in computational origami. In: Hull, T. (ed.) Origami\(\null^3\): Third International Meeting of Origami Science, Mathematics and Education, Natick, Massachusetts, pp. 3–16. A K Peters, Ltd (2002)Google Scholar
  5. 5.
    Geretschläger, R.: Geometric Constructions in Origami. Morikita Publishing Co. (2002) (In Japanese, translation by Fukagawa, Hidetoshi)Google Scholar
  6. 6.
    Hull, T.: Origami and geometric constructions (2005), http://www.merrimack.edu/~thull/omfiles/geoconst.html
  7. 7.
    Huzita, H.: Axiomatic Development of Origami Geometry. In: Huzita, H. (ed.) Proceedings of the First International Meeting of Origami Science and Technology, pp. 143–158Google Scholar
  8. 8.
    Ida, T., Ţepeneu, D., Buchberger, B., Robu, J.: Proving and Constraint Solving in Computational Origami. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS, vol. 3249, pp. 132–142. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Ida, T., Takahashi, H., Ţepeneu, D., Marin, M.: Morley’s Theorem Revisited through Computational Origami. In: Proceedings of the 7th International Mathematica Symposium (IMS 2005) (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Tetsuo Ida
    • 1
  • Hidekazu Takahashi
    • 1
  • Mircea Marin
    • 1
  • Fadoua Ghourabi
    • 1
  • Asem Kasem
    • 1
  1. 1.Department of Computer ScienceUniversity of TsukubaTsukubaJapan

Personalised recommendations