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Fundamentals of Fractal Sets, Space-Filling Curves and Their Applications in Electronics and Communications

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Intelligent Computing Based on Chaos

Part of the book series: Studies in Computational Intelligence ((SCI,volume 184))

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Summary

Geometric objects possessing properties impossible to describe using Euclidean notion of dimensionality are wide-spread in nature and are also encountered in many scientific experiments. Mathematical description of such objects has been focus of research for very long time - probably starting with the works of Georg Cantor, through von Koch to Julia/Fatou and Sierpinski just to name the most important contributors. The notion of fractal was coined by B. Mandelbrot and it is used for description of structures having non-integer dimension. Fractal geometric objects have several intriguing properties apart from its non-integer dimension, namely they can have finite area while showing infinite perimeter or infinite area for a finite volume object. They show also the self-similarity property - similar fine structure observed at any magnification scale. We discuss also the concept of space-filling curves introduced by Peano and Hilbert providing another type of geometric constructions having no fractal dimensionality but preserving the infinite length property on a finite area. These fundamental properties of fractal objects and space-filling curves can found very interesting applications in electrical and electronic engineering. We present some of the most spectacular of these applications: 1). fabrication of very large capacitances thanks to technological possibilities of making huge conducting areas in a limited volume; 2). enhancement of attainable capacitance values in IC design thanks to usage of lateral capacitances obtained by fractioning the available chip area; 3). Fabrication of multi-band antennas with improved impedance matching in a very small volume exploiting the self-similar properties of meandering structures and packaging of very long wires in a small volume.

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  1. Department of Information Technologies Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Reymonta 4, 30-059, Kraków, Poland

    Maciej J. Ogorzałek

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  1. Maciej J. Ogorzałek
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  1. University of California, La Jolla, CA, USA

    Ljupco Kocarev

  2. AGH University of Science and Technology , Krakow, Poland

    Zbigniew Galias

  3. France Telecom R&D Beijing, Beijing, China

    Shiguo Lian

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Ogorzałek, M.J. (2009). Fundamentals of Fractal Sets, Space-Filling Curves and Their Applications in Electronics and Communications. In: Kocarev, L., Galias, Z., Lian, S. (eds) Intelligent Computing Based on Chaos. Studies in Computational Intelligence, vol 184. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95972-4_3

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