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1 Journal of Applied Mathematics and Computational Mechanics

1 2015

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On a visualization of the convergence of the boundary of generalized Mandelbrot set to (n-1)-sphere

Katunin A., Fedio K.

Journal of Applied Mathematics and Computational Mechanics

2015

Vol. 14, nr 1

63--69

In this article we analyze the generalized Mandelbrot set in higher-order hypercomplex number spaces following both the Cayley-Dickson construction algebraic spaces and the spaces defined by Clifford algebras. The particular case of the generalized 3D quasi-Mandelbrot set was also considered. In particular, we investigated the increase of power of the iterated variable and proved that when this power tends to infinity, the Mandelbrot set is convergent to the unit circle. The same is true for the generalized Mandelbrot sets in higher-dimensional hypercomplex number spaces, i.e. when the power of iterated variable tends to infinity, the generalized Mandelbrot set is convergent to the unit (n-1)- sphere. The results of our investigation were visualized for the generalized Mandelbrot set in a complex number space and the generalized quasi-Mandelbrot set in a 3D Euclidean space.