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Curves of infinite length in labyrinth fractals

Part of: Classical measure theory Computational aspects in algebraic geometry Real and complex geometry Functions of several variables General convexity

Published online by Cambridge University Press:  30 March 2011

Ligia L. Cristea  and
Bertran Steinsky
Ligia L. Cristea
Affiliation:
Technical University of Graz, Steyrergasse 30, 8010 Graz, Austria (strublistea@gmail.com)
Bertran Steinsky
Affiliation:
Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria (bertran.steinsky@jku.at)
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Abstract

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We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

Keywords

infinite curve lengthfractaldendritetreeinfinite distance

MSC classification

Secondary: 14Q05: Curves 26B05: Continuity and differentiation questions 28A75: Length, area, volume, other geometric measure theory 28A80: Fractals 51M25: Length, area and volume 52A38: Length, area, volume
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 329 - 344
Copyright
Copyright © Edinburgh Mathematical Society 2011

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