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The Complex Geometry of the Mandelbrot Set

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ISCS 2013: Interdisciplinary Symposium on Complex Systems

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 8))

Abstract

In this paper, we give a brief overview of the geometry of the Mandelbrot set. We show how to distinguish each of the principal bulbs hanging off the main cardioid of this set by counting the spokes of the antennas attached to each bulb. We also use these antennas to attach a fraction to each such bulb, and this then indicates how these bulbs are arranged around the boundary of the main cardioid.

2000 MSC number: Primary 37F10; Secondary 37F45

This work was partially supported by grant #208780 from the Simons Foundation.

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References

  1. Devaney, R.L., Moreno Rocha, M.: Geometry of the antennas in the Mandelbrot set. Fractals 10, 39–46 (2002)

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  2. Devaney, R.L., Moreno Rocha, M.: The fractal geometry of the Mandelbrot set: I. periods of the bulbs. In: Fractals, Graphics, and Mathematics Education. MAA Notes, vol. 58, pp. 61–68 (2002)

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Authors and Affiliations

  1. Department of Mathematics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Robert L. Devaney

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  1. Robert L. Devaney
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Correspondence to Robert L. Devaney .

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Editors and Affiliations

  1. University of Tübingen, Tübingen, Germany

    Ali Sanayei

  2. Department of Computer Science, VÅ B-Technical University of Ostrava, Ostrava, Czech Republic

    Ivan Zelinka

  3. Institute of Physical and Theoretical Chemistry, University of Tübingen, Tübingen, Germany

    Otto E. Rössler

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Devaney, R.L. (2014). The Complex Geometry of the Mandelbrot Set. In: Sanayei, A., Zelinka, I., Rössler, O. (eds) ISCS 2013: Interdisciplinary Symposium on Complex Systems. Emergence, Complexity and Computation, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45438-7_1

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  • DOI: https://doi.org/10.1007/978-3-642-45438-7_1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45437-0

  • Online ISBN: 978-3-642-45438-7

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