Skip to main content

Abstract

This paper on the geometry, algebra and arithmetics of continued fractions is based on a lecture for students, teachers and a non-specialist audience, beginning with the history of the golden number and Fibonacci sequence, continued fractions of rational and irrational numbers, Lagrange theorem on periodicity of continued fractions for quadratic irrationals, Klein’s geometric interpretation of the convergents as integer points, Jung-Hirzebruch continued fractions with negative signs and two dimensional singularities, higher dimensional generalizations, and ending with a result on a periodic generalized 3-dimensional continued fraction for a cubic irrational.

Para Antonio Campillo en sus 65 vueltas alrededor del sol

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I.: Higher dimensional continued fractions. Regul. Chaotic Dyn 3, 10–17 (1998)

    Article  MathSciNet  Google Scholar 

  2. Beskin, N.: Fracciones Maravillosas. Ediciones MIR, Moscú (1987)

    Google Scholar 

  3. Coxeter, H.S.M.: Regular Polytopes. Dover Publications, New York (1973)

    MATH  Google Scholar 

  4. Coxeter, H.S.M.: The role of intermediate convergents in Tait’s explanation for phyllotaxis. J. Algebra 20, 167–175 (1972)

    Article  MathSciNet  Google Scholar 

  5. Euclid: The Thirteen Books of the Elements (translated by Sir T. L. Heath). Dover Publications, New York (1956)

    Google Scholar 

  6. Fibonacci: Liber Abaci. Springer, New York (2002)

    Google Scholar 

  7. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

    Google Scholar 

  8. Fowler, D.H.: The Mathematics of Plato’s Academy. A New Reconstruction, 2nd ed. Clarendon Press, Oxford (1999)

    Google Scholar 

  9. Giraud, J.: Surfaces d’Hilbert-Blumenthal (d’après Hirzebruch, etc …). In: Séminaire de Géométrie Algébrique d’Orsay, LNM 868. Springer, Berlin (1981)

    Google Scholar 

  10. Gonzalez Sprinberg, G.: Éventails en dimension deux et transformé de Nash. Secrétariat Mathémathique de l’E.N.S., Paris (1977)

    MATH  Google Scholar 

  11. Hirzebruch, F.: Über vierdimensionale Riemannsche Flächen Mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen. Math. Ann. 126, 1–22 (1953)

    Article  MathSciNet  Google Scholar 

  12. Hirzebruch, F.: Hilbert Modular Surfaces, vol. 19, no. 2, pp. 183–281. Enseignement Mathématique, Genève (1973)

    Google Scholar 

  13. Jung, H.W.E.: Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer Stelle x = a, y = b. J. Reine Angew. Math. 133, 289–314 (1908)

    Google Scholar 

  14. Klein, F.: Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung. Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl. 3, 357–359 (1895). French translation: Sur une représentation géométrique du développement en fraction continue ordinaire. Nouvelles Annales de Mathématiques 15, 327–331 (1896)

    Google Scholar 

  15. Khinchin, A.Ya.: Continued Fractions. Dover Publications, New York (1997)

    Google Scholar 

  16. Lipman, J.: Introduction to Resolution of Singularities. Proceedings of Symposia in Pure Mathematics, vol. 29, pp. 187–230. American Mathematical Society, Providence (1975)

    Google Scholar 

  17. Perron, O.: Die Lehre von den Kettenbrüchen. Teubner, Germany (1954)

    MATH  Google Scholar 

  18. Popescu-Pampu, P.: The geometry of continued fractions and the topology of surface singularities. In: Brasselet, J.-P., Suwa, T. (eds.) Singularities in Geometry and Topology 2004. Advanced Studies in Pure Mathematics, vol. 46, pp. 119–195. Mathematical Society of Japan, Tokyo (2007)

    MATH  Google Scholar 

  19. Stark, H.M.: An Introduction to Number Theory. Markham Publishing Company, Chicago (1970); MIT Press, Cambridge (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerardo Gonzalez Sprinberg .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gonzalez Sprinberg, G. (2018). On Continued Fractions. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_27

Download citation

Publish with us

Policies and ethics