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Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set

Author

Listed:
  • Young-Hee Geum

    (Department of Mathematics, Dankook University, Cheonan 330-714, Korea)

  • Young-Ik Kim

    (Department of Mathematics, Dankook University, Cheonan 330-714, Korea)

Abstract

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical properties. The budding point of the period- 3 k component, which is born on the boundary of the period-3 component, and its relevant period- 3 k points are theoretically obtained by means of Cardano’s formula for the cubic equation. In addition, computational results are presented in tables and figures to support the theoretical background of this paper.

Suggested Citation

  • Young-Hee Geum & Young-Ik Kim, 2021. "Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set," Mathematics, MDPI, vol. 9(19), pages 1-15, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2519-:d:651326
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    References listed on IDEAS

    as
    1. Young Ik Kim & Young Hee Geum, 2013. "A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, February.
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