Abstract
Construction of geometrical objects by origami, the Japanese traditional art of paper folding, is enjoyable and intriguing. It attracted the minds of artists, mathematicians and computer scientists for many centuries. Origami will become a more rigorous, effective and enjoyable art if the origami constructions can be visualized on the computer and the correctness of the constructions can be automatically proved by an algorithm. We call the methodology of visualizing and automatically proving origami constructions computational origami. As a non-trivial example, in this paper, we visualize a construction of a regular heptagon by origami and automatically prove the correctness of the construction.
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References
Alperin, R.C.: A Mathematical Theory of Origami Constructions and Numbers. New York J. Math. 6, 119–133 (2000)
Buchberger, B.: Gröbner-Bases: An Algorithmic Method in Polynomial Ideal Theory. In: Bose, N.K. (ed.) Multidimensional Systems Theory - Progress, Directions and Open Problems in Multidimensional Systems, Ch. 6. Reidel Publishing Company, Dodrecht (1985) (2nd edn. Kluwer Academic Publisher 2003)
Buchberger, B., Jebelean, T., Kriftner, F., Marin, M., Tomuta, E., Vasaru, D.: An overview on the Theorema project. In: Kuechlin, W. (ed.) Procdings of ISSAC 1997 (International Symposium on Symbolic and Algebraic Computation, Maui, Hawaii, July 21-23. ACM Press, New York (1997)
Buchberger, B., Dupre, C., Jebelean, T., Kriftner, F., Nakagawa, K., Vasaru, D., Windsteiger, W.: The Theorema Project: A Progress Report. In: Kerber, M., Kohlhase, M. (eds.) Symbolic Computation and Automated Reasoning: The Calculemus-2000 Symposium (Symposium on the Integration of Symbolic Computation and Mechanized Reasoning (August 6-7, 2000); A K Peters Ltd., St. Andrews, Scotland (2001)
Buchberger, B.: Towards the Automated Synthesis of a Groebner bases Algorithm. RACSAM, Reviews of the Spanish Royal Academy of Science. Serie A: Mathematicas 98(1), 65–75 (2004)
Chou, S.-C.: Mechanical geometry theorem proving. Reidel, Dordrecht Boston (1988)
Geretschläger, R.: Euclidean constructions and the geometry of origami. Math. Mag. 68(5), 357–371 (1995)
Gleason, A.M.: Angle Trisection, the Heptagon, and the Triskaidecagon. American Mathematical Monthly 95(3), 185–194 (1998)
Huzita, H.: Axiomatic Development of Origami Geometry. In: Proceedings of the First International Meeting of Origami Science and Technology, pp. 143–158 (1989)
Huzita, H.: Drawing the regular heptagon and the regular nonagon by origami (paper folding). Symmetry: Culture and Science 5(1), 69–84 (1994)
Ida, T., Ţepeneu, D., Buchberger, B., Robu, J.: Proving and Constraint Solving in Computational Origami. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 132–142. Springer, Heidelberg (2004)
Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symbolic Computation 2(4), 399–408 (1986)
Kutzler, B., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput. 2, 389–397 (1986)
Robu, J.: Automated Geometric Theorem Proving (PhD Thesis), RISC-Linz Report Series No. 02-23. Johannes Kepler University Linz, Austria (2002)
Ţepeneu, D., Ida, T.: MathGridLink - A bridge between Mathematica and “the Grid”. In: The 20th Annual Conference of Japan Society of Software Science and Technology, Nagoya (September 2003)
Wu, W.-T.: Basic principles of mechanical theorem proving in elementary geometries. J. Automat. Reason. 2, 221–252 (1986)
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Robu, J., Ida, T., Ţepeneu, D., Takahashi, H., Buchberger, B. (2006). Computational Origami Construction of a Regular Heptagon with Automated Proof of Its Correctness. In: Hong, H., Wang, D. (eds) Automated Deduction in Geometry. ADG 2004. Lecture Notes in Computer Science(), vol 3763. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11615798_2
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DOI: https://doi.org/10.1007/11615798_2
Publisher Name: Springer, Berlin, Heidelberg
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