Research ArticleNo Access

3D RENDERING OF THE QUATERNION MANDELBROT SET WITH MEMORY

RICARDO FARIELLO

https://orcid.org/0000-0002-5868-4068

Universidade Estadual de Montes Claros, Montes Claros, MG, Brazil

E-mail Address: [email protected]

Corresponding author.

Search for more papers by this author

PAUL BOURKE

https://orcid.org/0000-0002-0325-882X

The University of Western Australia, Perth, Australia

Search for more papers by this author

, and

GABRIEL V. S. ABREU

https://orcid.org/0009-0005-5916-3271

Universidade Estadual de Montes Claros, Montes Claros, MG, Brazil

Search for more papers by this author

https://doi.org/10.1142/S0218348X24500610Cited by:0 (Source: Crossref)

Abstract

In this paper, we explore the quaternion equivalent of the Mandelbrot set equipped with memory and apply various visualization techniques to the resulting $4$ -dimensional geometry. Three memory functions have been considered, two that apply a weighted sum to only the previous two terms and one that performs a weighted sum of all previous terms of the series. The visualization includes one or two cutting planes for dimensional reduction to either $3$ or $2$ dimensions, respectively, as well as employing an intersection with a half space to trim the $3$ D solids in order to reveal the interiors. Using various metrics, we quantify the effect of each memory function on the structure of the quaternion Mandelbrot set.

Keywords:

References

1. G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya , On periodic and chaotic regions in the Mandelbrot set, Chaos, Solitons Fractals 32(1) (2007) 15–25. Crossref, Web of Science, Google Scholar
2. R. Rojas , A tutorial on efficient computer graphic representations of the Mandelbrot set, Comput. Graph. 15(1) (1991) 91–100. Crossref, Web of Science, Google Scholar
3. P. Petek , On the quaternionic Julia sets, in Chaotic Dynamics: Theory and Practice, ed. T. Bountis (Springer US, Boston, MA, 1992), pp. 53–58. Crossref, Google Scholar
4. J. Gomatam, J. Doyle, B. Steves and I. McFarlane , Generalization of the Mandelbrot set: Quaternionic quadratic maps, Chaos, Solitons Fractals 5(6) (1995) 971–986. Crossref, Web of Science, Google Scholar
5. J. Gomatam, J. Doyle and B. Steves , Quaternionic generalisation of the Mandelbrot set, in From Newton to Chaos: Modern Techniques for Understanding and Coping with Chaos in N-Body Dynamical Systems, eds. A. E. Roy and B. A. Steves (Springer US, Boston, MA, 1995), pp. 557–562. Crossref, Google Scholar
6. S. Bedding and K. Briggs , Iteration of quaternion functions, Am. Math. Monthly 103(8) (1996) 654–664. Crossref, Web of Science, Google Scholar
7. S. Nakane , Dynamics of a family of quadratic maps in the quaternion space, Int. J. Bifurc. Chaos 15(8) (2005) 2535–2543. Link, Web of Science, Google Scholar
8. C. A. Pickover , Visualization of quaternion slices, Image Vis. Comput. 6(4) (1988) 235–237. Crossref, Web of Science, Google Scholar
9. A. Norton , Generation and display of geometric fractals in 3-D, in Proceedings of the 9th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH’82 (Association for Computing Machinery, New York, NY, USA, 1982), pp. 61–67. Crossref, Google Scholar
10. A. Norton , Julia sets in the quaternions, Comput. Graph. 13(2) (1989) 267–278. Crossref, Web of Science, Google Scholar
11. J. A. R. Holbrook , Quaternionic astroids and starfields, Appl. Math. Notes 8(2) (1983) 1–34. Google Scholar
12. J. A. R. Holbrook , Quaternionic Fatou–Julia sets, Ann. Sci. Math. Québec 11(1) (1987) 79–94. Google Scholar
13. M. Abbas, H. Iqbal and M. De la Sen , Generation of Julia and Mandelbrot sets via fixed points, Symmetry 12(1) (2020) 86. Crossref, Web of Science, Google Scholar
14. C. Zou, A. A. Shahid, A. Tassaddiq, A. Khan and M. Ahmad , Mandelbrot sets and Julia sets in Picard–Mann orbit, IEEE Access 8 (2020) 64411–64421. Crossref, Web of Science, Google Scholar
15. A. A. Shahid, W. Nazeer and K. Gdawiec , The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets, Monatsh. Math. 195(4) (2021) 565–584. Crossref, Web of Science, Google Scholar
16. S. Kumari, K. Gdawiec, A. Nandal, M. Postolache and R. Chugh , A novel approach to generate Mandelbrot sets, Julia sets and biomorphs via viscosity approximation method, Chaos, Solitons Fractals 163 (2022) 112540. Crossref, Web of Science, Google Scholar
17. S. Kumari, K. Gdawiec, A. Nandal, N. Kumar and R. Chugh , An application of viscosity approximation type iterative method in the generation of Mandelbrot and Julia fractals, Aequ. Math. 97(2) (2023) 257–278. Crossref, Web of Science, Google Scholar
18. C. Beck , Physical meaning for Mandelbrot and Julia sets, Physica D 125(3) (1999) 171–182. Crossref, Web of Science, Google Scholar
19. R. Alonso-Sanz , On quaternion maps with memory, Complex Syst. 24(3) (2015) 223–232. Crossref, Google Scholar
20. Y. Sun, P. Li and Z. Lu , Generalized quaternion M sets and Julia sets perturbed by dynamical noises, Nonlinear Dyn. 82(1) (2015) 143–156. Crossref, Web of Science, Google Scholar
21. R. Alonso-Sanz , A glimpse of complex maps with memory, Complex Syst. 21(4) (2013) 269–282. Crossref, Google Scholar
22. R. Alonso-Sanz , The Mandelbrot set with partial memory, Complex Syst. 23(3) (2014) 227–238. Crossref, Google Scholar
23. R. Alonso-Sanz , On complex maps with delay memory, Fractals 23(3) (2015) 1550027. Link, Web of Science, Google Scholar
24. A. Fulinski and A. S. Kleczkowski , Nonlinear maps with memory, Phys. Scr. 35(2) (1987) 119. Crossref, Web of Science, Google Scholar
25. R. Alonso-Sanz , Discrete Systems with Memory (World Scientific, 2011). Link, Google Scholar
26. J. C. Hart, D. J. Sandin and L. H. Kauffman , Ray tracing deterministic 3-D fractals, ACM SIGGRAPH Comput. Graph. 23(3) (1989) 289–296. Crossref, Google Scholar
27. V. da Silva, T. Novello, H. Lopes and L. Velho , Real-time rendering of complex fractals, in Ray Tracing Gems II: Next Generation Real-Time Rendering with DXR, Vulkan, and OptiX, eds. A. Marrs, P. Shirley and I. Wald (Apress, Berkeley, CA, 2021), pp. 529–544. Crossref, Google Scholar
28. Q. Zhang, R. Eagleson and T. M. Peters , Volume visualization: A technical overview with a focus on medical applications, J. Digital Imag. 24 (2011) 640–664. Crossref, Web of Science, Google Scholar
29. S. Fidan , The use of micro-CT in materials science and aerospace engineering, in Micro-Computed Tomography (micro-CT) in Medicine and Engineering, ed. K. Orhan (Springer International Publishing, Cham, 2020), pp. 267–276. Crossref, Google Scholar
30. C. Ma and J. Rokne , 3D seismic volume visualization, in Integrated Image and Graphics Technologies, eds. D. D. Zhang, M. Kamel and G. Baciu (Springer US, Boston, MA, 2004), pp. 241–262. Crossref, Google Scholar
31. A. Limaye , Drishti: A volume exploration and presentation tool, in Developments in X-Ray Tomography VIII, Vol. 8506, ed. S. R. Stock (International Society for Optics and Photonics, SPIE, 2012), p. 85060X. Crossref, Google Scholar
32. P. Bourke , Visualising volumetric fractals, GSTF Int. J. Comput. 5(2) (2017) 58–62. Google Scholar
33. R. Fariello, P. Bourke and J. P. Lopes , Calculating Julia fractal sets in any embedding dimension, Fractals, 31(1) (2023) 2350018. Link, Web of Science, Google Scholar