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Journal for Geometry and Graphics 22 (2018), No. 1, 049--058
Copyright Heldermann Verlag 2018

The Generalized Biquaternionic M-J Sets

Andrzej Katunin
Inst. of Fundamentals of Machinery Design, Silesian University of Technology, Konarskiego 18A, 44-100 Gliwice, Poland

The Mandelbrot-Julia sets, henceforth abbreviated as M-J sets, and their properties have been extensively studied since their discovery. Many studies are focused on properties and dynamics of generalized M-J sets in complex and hypercomplex vector spaces, however there are still many variations of M-J sets which have not been studied yet. The following paper discusses one of such variations -- the M-J sets in the biquaternionic vector space. Starting from theoretical fundamentals on an algebra of biquaternions and its closedness under addition and multiplication, the author defines the generalized biquaternionic M-J sets and their relation both with complex M-J sets as well as with their 4-space analogues: quaternionic and bicomplex M-J sets. The connectedness and dynamics of J sets is also studied. Moreover, the analysis of 3D cross-sections of J sets allows validating the relationships with other hypercomplex fractal sets and evaluating a symmetry of resulting biquaternionic sets.

Keywords: Biquaternionic Mandelbrot-Julia sets, algebra of biquaternions, fractals, generalized Mandelbrot-Julia sets.

MSC: 28A80

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