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

Young-Hee Geum and Young-Ik Kim
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Young-Hee Geum: Department of Mathematics, Dankook University, Cheonan 330-714, Korea
Young-Ik Kim: Department of Mathematics, Dankook University, Cheonan 330-714, Korea

Mathematics, 2021, vol. 9, issue 19, 1-15

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.

Keywords: parameter space; Mandelbrot set; budding point; hyperbolic component; critical orbit (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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