Skip to main content
SpringerLink
Log in

A characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

It is well-known that the side length of a regular hexagon is half the length of its longest diagonals. From this property, one can easily see that for every positive integer \(m>1\), any regular 6m-gon contains two non-congruent diagonals that are commensurable. In this paper, we show that if n is not a multiple of 6, then all pairs of diagonals of different lengths of a regular n-gon are incommensurable. This yields a characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable. The main result gives positive answers to some questions on this topic.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Anatriello, G., Laudano, F., Vincenzi, G.: Sur l’ incommensurabilité dans les polygones réguliers. Quadrature 115, 1–5 (2020)

    Google Scholar 

  2. Burton, D.: Elementary Number Theory. McGraw-Hill, New York (2012)

    Google Scholar 

  3. Barbeau, E.J.: Incommensurability proof: a pattern that peters out. Math Mag. 56(2), 82–90 (1983)

    Article  MathSciNet  Google Scholar 

  4. Becker, L.C., Becker, C.B.: Encyclopedia of Ethics. Routledge, New York (2013)

    Book  Google Scholar 

  5. Filep, L.: Proportion theory in Greek mathematics. Acta Math. Acad. Paedaog. Nyházi. (N.S.). 19(2), 167–174 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Fontaine, A., Hurley, S.: Proof by picture: products and reciprocals of diagonal length ratios in the regular polygon. Forum Geom. 6, 97–101 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Havil, J.: The Irrationals. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  8. Harborth, H.: Diagonalen im regularen \(n\)-Eck. Elem. Math. 24, 104–109 (1969)

    MathSciNet  MATH  Google Scholar 

  9. Heineken, H.: Regelmässige Vielecke und ihre Diagonalen II. Rend. Sem. Mat. Univ. Padova 41, 332–344 (1968)

    MathSciNet  MATH  Google Scholar 

  10. Kepler, J.: Harmonices Mundi. In: Linz L.V. (ed.) anastatica: Brussels, Culture et civilisation, 1968; Bologna, Forni (1969)

  11. Lagerlund, H.: Encyclopedia of Medieval Philosophy: Philosophy Between 500 and 1500. Springer, Berlin (2010)

    Google Scholar 

  12. Niven, I.: Irrational Numbers. The Carus Mathematical Monograph. The Mathematical Association of America. Wiley, Hoboken (1956)

  13. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)

    MATH  Google Scholar 

  14. Rigby, J.F.: Multiple intersections of diagonals of regular polygons, and related topics. Geom. Dedicata 9, 207–238 (1980)

    Article  MathSciNet  Google Scholar 

  15. Stillwell, J.: Numbers and Geometry. UTM, Springer (1998)

  16. Von Fritz, K.: The discovery of incommensurability by hippasus of metapontum. Ann. Math. Second Ser. 46(2), 242–264 (1945)

    Article  MathSciNet  Google Scholar 

  17. Xiao, J.: Elementary Number Theory. International Press of Boston, Boston (2010)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA - INdAM, Ministry of Italian Research).

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Giovanni Vincenzi

Authors
  1. Giovanni Vincenzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Vincenzi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vincenzi, G. A characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable. Arch. Math. 115, 467–477 (2020). https://doi.org/10.1007/s00013-020-01477-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-020-01477-w

Keywords

Mathematics Subject Classification

Search

Navigation