Skip to main content
SpringerLink
Log in

Curves of infinite length in 4 × 4-labyrinth fractals

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.

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. Allouche J.-P., Shallit J.: Automatic Sequences: Theory, Applications, Generalizations, pp. 155–156. Cambridge University Press, Cambridge England (2003)

    MATH  Google Scholar 

  2. Cristea, L.L.: On the connectedness of limit net sets. Submitted (2008)

  3. Falconer K.J.: Fractal Geometry, Mathematical Foundations and Applications. John & Wiley Sons, Chichester (1990)

    MATH  Google Scholar 

  4. Hilbert D.: Über die stetige Abbildung einer Linie auf ein Flachenstück. Math. Ann. 38, 459–460 (1891)

    Article  MathSciNet  Google Scholar 

  5. Kuratowski K.: Topology. vol. 2. Academic Press, New York and London (1968)

    Google Scholar 

  6. Moran P.A.P.: Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42, 15–23 (1946)

    Article  MATH  MathSciNet  Google Scholar 

  7. Peano G.: Sur une courbe, qui remplit une aire plane. Math. Ann. 36, 157–160 (1890)

    Article  MathSciNet  Google Scholar 

  8. Pesin Y., Weiss H.: On the dimension of deterministic and random cantor-like sets. Comm. Math. Phys. 182, 105–153 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Pesin Y., Weiss H.: A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86, 233–275 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Seneta, E.: Non-negative matrices and Markov chains, 2nd edn. Springer Verlag, New York (1973, 1981, 2006)

  11. Tricot C.: Curves and Fractal Dimension. Springer-Verlag, Paris (1993)

    MATH  Google Scholar 

  12. von Koch H.: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Archiv för Matemat. 1, 681–702 (1904)

    Google Scholar 

  13. von Koch H.: Une méthode géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planes. Acta Math. 30, 145–174 (1906)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University of Graz, Steyrergasse 30, 8010, Graz, Austria

    Ligia Loreta Cristea & Bertran Steinsky

Authors
  1. Ligia Loreta Cristea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bertran Steinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bertran Steinsky.

Additional information

The authors are supported by the Austrian Science Fund (FWF), Project P-20412.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cristea, L.L., Steinsky, B. Curves of infinite length in 4 × 4-labyrinth fractals. Geom Dedicata 141, 1–17 (2009). https://doi.org/10.1007/s10711-008-9340-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-008-9340-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation