Abstract
All possible 4D hypercomplex vector spaces were considered in the light of an ability of construction of Julia fractals in them. Both arithmetic fundamentals of the considered algebras as well as implementation procedures of such hypercomplex numbers are given. In the paper, the presented study summarizes well-known 4D hypecomplex fractals, like bicomplex and quaternionic ones, introduces a group of new hypercomplex fractals, like biquaternionic, and shows why other 4D hypercomplex vector spaces cannot produce the non-trivial Julia sets. All of the considered cases were enriched by several graphical representations of hypercomplex Julia sets with their graphical analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bogush, A.A., Gazizov, A.Z., Kurochkin, Y.A., Stosui, V.T.: Symmetry properties of quaternionic and biquaterionic analogs of Julia sets. Ukrainian J. Phys. 48(4), 295–299 (2003)
Gintz, T.W.: Artist’s statement CQUATS - a non-distributive quad algebra for 3D renderings of Mandelbrot and Julia sets. Comput. Graph. 26(2), 367–370 (2002)
Hart, J.C., Sandin, D.J., Kauffman, L.H.: Ray tracing deterministic 3-D fractals. Comput. Graph. 23(3), 289–296 (1989)
Holbrook, J.A.R.: Quaternionic Fatou-Julia sets. Ann. Sci. Math Que. 11, 79–94 (1987)
Jafari, M.: Split semi-quaternions algebra in semi-euclidean 4-space. Cumhur. Sci. J. 36(1), 70–77 (2015)
Jafari, M., Yayli, T.: Generalized quaternions and their algebraic properties. Sér. A1. Math. Stat. 64(1), 15–27 (2015). Communications de la Faculté des Sciences de l’Université d’Ankara
Katunin, A.: The generalized biquaternionic M-J sets. Fractals, submitted (2016)
Katunin, A.: On the convergence of multicomplex M-J sets to the Steinmetz hypersolids. J. Appl. Math. Comput. Mech. 15(3) (in press, 2016)
Katunin, A., Fedio, K.: On a visualization of the convergence of the boundary of generalized Mandelbrot set to \((n-1)\)-sphere. J. Appl. Math. Comput. Mech. 14(1), 63–69 (2015)
Mortazaasl, H., Jafari, M.: A study on semi-quaternions algebra in semi-euclidean 4-space. Math. Sci. Appl. E-notes 1(2), 20–27 (2013)
Norton, A.V.: Generation and display of geometric fractals in 3-D. Comput. Graph. 16(3), 61–67 (1982)
Norton, A.V.: Julia sets in the quaternions. Comput. Graph. 13(2), 267–278 (1989)
Rochon, D.: A generalized Mandelbrot set for bicomplex numbers. Fractals 8(4), 355–368 (2000)
Rosenfeld, B.: Geometry of Lie Groups, Mathematics and Its Applications, vol. 393. Springer, Dordrecht (1997)
Wang, X.Y., Song, W.J.: The generalized M-J sets for bicomplex numbers. Nonlinear Dyn. 72(1), 17–26 (2013)
Wang, X.Y., Sun, Y.Y.: The general quaternionic M-J sets on the mapping \(z\leftarrow z^{\alpha }+c\) \((\alpha \in \mathbf{N})\). Comput. Math. Appl. 53(11), 1718–1732 (2007)
Zireh, A.: A generalized Mandelbrot set of polynomials of type \(e_{d}\) for bicomplex numbers. Georgian Math. J. 15(1), 189–194 (2008)
Acknowledgements
The publication is financed from the statutory funds of the Faculty of Mechanical Engineering of the Silesian University of Technology in 2016.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Katunin, A. (2016). Analysis of 4D Hypercomplex Generalizations of Julia Sets. In: Chmielewski, L., Datta, A., Kozera, R., Wojciechowski, K. (eds) Computer Vision and Graphics. ICCVG 2016. Lecture Notes in Computer Science(), vol 9972. Springer, Cham. https://doi.org/10.1007/978-3-319-46418-3_56
Download citation
DOI: https://doi.org/10.1007/978-3-319-46418-3_56
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46417-6
Online ISBN: 978-3-319-46418-3
eBook Packages: Computer ScienceComputer Science (R0)