Skip to main content
SpringerLink
Log in

Minimal Peano curve

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

A Peano curve p(x) with maximum square-to-linear ratio |p(x)−p(y)|2/|x−y| equal to 5 2/3 is constructed; this ratio is smaller than that of the classical Peano-Hilbert curve, whose maximum square-to-linear ratio is 6. The curve constructed is of fractal genus 9 (i.e., it is decomposed into nine fragments that are similar to the whole curve) and of diagonal type (i.e., it intersects a square starting from one corner and ending at the opposite corner). It is proved that this curve is a unique (up to isometry) regular diagonal Peano curve of fractal genus 9 whose maximum square-to-linear ratio is less than 6. A theory is developed that allows one to find the maximum square-to-linear ratio of a regular Peano curve on the basis of computer calculations.

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

Similar content being viewed by others

References

  1. E. V. Shchepin, “Dimension-Raising Maps and Continuous Information Transmission,” in Problems in Pure and Applied Mathematics (Priokskoe Knizhnoe Izdatel’stvo, Tula, 1987), Vol. 1, pp. 148–155 [in Russian].

    Google Scholar 

  2. E. V. Shchepin, “On Fractal Peano Curves,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 247, 294–303 (2004) [Proc. Steklov Inst. Math. 247, 272–280 (2004)].

    MathSciNet  Google Scholar 

  3. E. V. Shchepin and K. E. Bauman, “Peano Curves of Fractal Genus 9,” in Modelling and Data Analysis: Proceedings of the Faculty of Information Technologies, Moscow City Psychological-Pedagogical University (Rusavia, Moscow, 2004), Issue 1, pp. 79–89 [in Russian].

    Google Scholar 

  4. K. E. Bauman, “The Dilation Factor of the Peano-Hilbert Curve,” Mat. Zametki 80(5), 643–656 (2006) [Math. Notes 80, 609–620 (2006)].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Institute of Mathematics, Russian Academy of Science, ul. Gubkina 8, Moscow, 119991, Russia

    E. V. Shchepin

  2. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    K. E. Bauman

Authors
  1. E. V. Shchepin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. E. Bauman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © E.V. Shchepin, K.E. Bauman, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 251–271.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shchepin, E.V., Bauman, K.E. Minimal Peano curve. Proc. Steklov Inst. Math. 263, 236–256 (2008). https://doi.org/10.1134/S0081543808040172

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543808040172

Keywords

Search

Navigation