Abstract
By using a systematic, automated way, we discover a large amount of geometry statements on regular polygons. Given a regular n-gon, its diagonals are taken, two pairs of them may determine a pair of intersection points that define a segment. By considering all possible segments defined in this way, we can compute the lengths of them symbolically, and, depending on the simplicity of the symbolic result we classify the segment either as “interesting” or “not interesting”.
Among others, we prove that in a regular 11-gon with unit sides the only rational lengths appearing the way described above, are 1 and 2, and the only quadratic surd is \(\sqrt{3}\). The applied way of proving is exhaustion, by using the freely available software tool RegularNGons, programmed by the author. The combinatorial explosion, however, calls for future improvements involving methods in artificial intelligence.
The symbolic method being used is Wu’s algebraic geometry approach [1], combined with the discovery algorithm communicated by Recio and Vélez [2]. The heavy computations are performed by a recent version of the Giac computer algebra software, running in a web browser with the support of the recent technology WebAssembly. Visual communication of the obtained results is operated by the dynamic geometry software GeoGebra.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Wu, W.T.: On the decision problem and the mechanization of theorem-proving in elementary geometry (1984)
Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23, 63–82 (1999)
Colton, S., Bundy, A., Walsh, T.: On the notion of interestingness in automated mathematical discovery. Int. J. Hum.-Comput. Stud. 53, 351–375 (2000)
Chou, S.C.: Mechanical Geometry Theorem Proving. Springer, Dordrecht (1987)
Wantzel, P.: Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas. J. Math. Pures Appl. 1, 366–372 (1837)
Sethuraman, B.: Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra via Geometric Constructibility. Springer, New York (1997). https://doi.org/10.1007/978-1-4757-2700-5
Pierpont, J.: On an undemonstrated theorem of the disquisitiones arithmeticæ. Bull. Am. Math. Soc. 2, 77–83 (1895)
Gleason, A.M.: Angle trisection, the heptagon, and the triskaidecagon. Am. Math. Mon. 95, 185–194 (1988)
Watkins, W., Zeitlin, J.: The minimal polynomial of \(\cos (2\pi /n)\). Am. Math. Mon. 100, 471–474 (1993)
Lehmer, D.H.: A note on trigonometric algebraic numbers. Am. Math. Mon. 40, 165–166 (1933)
Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symb. Comput. 2, 399–408 (1986)
Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover Publications, New York (1973)
Kovács, Z.: RegularNGons. A GitHub project (2018). https://github.com/kovzol/RegularNGons
Haas, A., et al.: Bringing the web up to speed with WebAssembly. In: Proceedings of the 38th ACM SIGPLAN Conference on Programming Language Design and Implementation. Association for Computing Machinery, pp. 185–200 (2017)
Acknowledgements
The author was partially supported by a grant MTM2017-88796-P from the Spanish MINECO (Ministerio de Economia y Competitividad) and the ERDF (European Regional Development Fund). Many thanks to Tomás Recio and Francisco Botana for their valuable comments and suggestions. The JavaScript implementation in RegularNGons was supported by Gábor Ancsin and the GeoGebra Team.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Kovács, Z. (2018). Discovering Geometry Theorems in Regular Polygons. In: Fleuriot, J., Wang, D., Calmet, J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2018. Lecture Notes in Computer Science(), vol 11110. Springer, Cham. https://doi.org/10.1007/978-3-319-99957-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-99957-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99956-2
Online ISBN: 978-3-319-99957-9
eBook Packages: Computer ScienceComputer Science (R0)