Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

On the Mandelbrot Set for i^2 = ±1 and Imaginary Higgs Fields

Download PDF (2618.3 KB) PP. 27 - 54 Pub. Date: April 1, 2021

DOI: 10.22606/jaam.2021.62001


  • Jonathan Blackledge*
    Stokes Professor, Science Foundation Ireland; Distinguished Professor, Centre for Advanced Studies, Warsaw University of Technology, Poland; Visiting Professor, Faculty of Arts, Science and Technology, Wrexham Glyndwr University of Wales, UK; Professor Extraordinaire, Faculty of Natural Sciences, University of Western Cape, South Africa; Honorary Professor, School of Electrical and Electronic Engineering, Technological University Dublin, Ireland; Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa.


We consider the consequence of breaking with a fundamental result in complex analysis by letting i^2 = ±1 where i = √−1 is the basic unit of all imaginary numbers. An analysis of the Mandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object is possible based on i^2 = −1 and i^2 = +1, respectively. Further, we consider the transient behaviour associated with the two cases to produce a range of non-standard sets in which a Fractal geometric structure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclidean object is a square whose properties are investigate. Coupled with the associated Julia sets and other complex plane mappings, this approach provides the potential to generate a wide range of new semi-fractal structures which are visually interesting and may be of artistic merit. In this context, we present a mathematical paradox which explores the idea that i^2 = ±1. This is based on coupling a well known result of the Riemann zeta function (i.e. (0) = −1/2) with the Grandi’s series, both being examples of Ramanujan sums. We then explore the significance of this result in regard to an interpretation of the fundamental field equations of Quantum Mechanics when a Higgs field is taken to be produced by an imaginary mass im such that (±im)^2 = +m^2. A set of new field equations are derived and studied. This includes an evaluation of the propagators (the free space Green’s functions) which exhibit decay characteristics over very short (sub-atomic) distances.


non-standard Mandelbrot set, transient characteristics, imaginary mass, causal tachyons, Higgs fields


[1] I. Stewart and D. Tall, “Complex Analysis”, Edition 2, Cambridge University Press, 2018. ISBN: 978-1-43679-3

[2] A. Douady and J. H. Hubbard, “étude Dynamique des Polyn?mes cComplexes”, Société Mathématique de France, 2007. Available at:

[3] B. Mandelbrot, “The Fractal Geometry of Nature”, W. H. Freeman and Co., 1983. ISBN: 978-0-7167-1186-5

[4] Euclid, “Elements”, (Eds. D. Densmore and T. L. Heath), Green Lion Press, 2002. ISBN: 978-1-888009-19-4

[5] L. Carleson and T. W. Gamelin, “Complex Dynamics”, Springer 1993. ISBN: 978-0-387-97942-7

[6] A. F. Beardon, “Iteration of Rational Functions”, Springer 1991. ISBN: 0-387-95151-2

[7] H. M. Edwards, “Riemann’s Zeta Function”, Academic Press, 1974. ISBN-10: 0-486-41740-9 .

[8] G. T. Bagni, “Infinite Series from History to Mathematics Education”, International Journal for Mathematics Teaching and Learning, 2005. Available at:

[9] G. Feinberg, G. (1967). “Possibility of Faster-Than-Light Particles”. Physical Review, vol. 159, no. 5, pp. 1089?1105, 1967.

[10] W. Greiner, “Relativistic Quantum Mechanics: Wave Equations” (Edition 3), Springer Verlag, 2000, ISBN: 3-5406-74578.

[11] G. Bernardi, M. Carena and T. Junk, “Higgs bosons: Theory and searches”, Review: Hypothetical particles and Concepts. Particle Data Group, 2007. Available at:

[12] P. W. Higgs, “Broken Symmetries and the Masses of Gauge Bosons”, Physical Review Letters, vol. 13 , no. 16, pp. 508?09, 1964.

[13] G. Aad et al. “Combined Measurement of the Higgs Boson Mass in pp Collisions at p s= 7 and 8 TeV with the ATLAS and CMS Experiments”, Physical Review Letters, 114, 191803, 2015. Available at: https: //

[14] J. W. Milnor, “Dynamics in One Complex Variable”, Edition 3, Annals of Mathematics Studies 160, Princeton University Press 2006. Available at:

[15] E. Demidov, “The Mandelbrot and Julia sets Anatomy”, A virtual investigation with interactive pictures, 2003. Available at:

[16] M. Michelitsch and O. E. R?ssler, “The ‘Burning Ship’ and Its Quasi-Julia Sets”. In: Computers & Graphics, vol. 16, no. 4, pp. 435-438, 1992.

[17] MATLAB, Mathematical Computing Software, The MathWorks Inc., 2020. Available at: https://uk.

[18] B. C. Berndt, “Ramanujan’s Notebooks”, Ramanujan’s Theory of Divergent Series, Springer, 1985. Available at:

[19] R. S. Vieira, “An Introduction to the Theory of Tachyons”, Rev. Bras. Ens. Fis. vol. 34, no. 3, pp. 1-17, 2011. Available at

[20] J. M. Hill and B. J. Cox, “Einstein’s Special Relativity Beyond the Speed of lLight”, Proc. R. Soc. A vol. 468 (2148), pp. 4174?4192, 2012. Available at: 2012.0340

[21] J. R. Forshaw and A. G. Smith, “Dynamics and Relativity” Wiley, 2009. ISBN 978-0-470-01460-8

[22] D. McMahon, “Relativity. Demystified” McGraw-Hill, 2006. ISBN 0-07-145545-0

[23] N Boston, “The Proof of Fermat’s Last Theorem”, University of Wisconsin, 2003. Available at: https: //

[24] N. Zettili, “Quantum Mechanics: Concepts and Applications”, Edition 2, Wiley, 2009. ISBN: 978-0-470-02678-6

[25] O. Klein, “Quantentheorie und fünfdimensionale Relativit?tstheorie”, Zeitschrift für Physik, vol. 37, no. 12, pp. 895-906, 1926. Available at: pdf

[26] W. Gordon, “Der Comptoneffekt nach der Schr?dingerschen Theorie”, Zeitschrift für Physik, vol. 40, 117-133, 1926. Available from:

[27] S. Weinberg, “The Quantum Theory of Fields”, Cambridge University Press, 2002. ISBN 0-521-55001-7

[28] E. Schr?dinger, “An Undulatory Theory of the Mechanics of Atoms and Molecules” Physical Review. vol. 28 no. 6, pp. 1049?1070, 2026. Available at: //

[29] E. Schr?dinger, “Quantization as an eigenvalue problem,” Annalen der Physik, vol. 489, no. 79, 1926.

[30] J. M. Blackledge and B. Babajanov, “The Fractional Schr?dinger-Klein-Gordon Equation and Intermediate Relativism”, Mathematica Eterna, vol. 3, no. 8, 601-615, 2013. Available from: articles/the-fractional-schrodingerkleingordon-equation-and-intermediate-relativism.pdf

[31] P. A. M. Dirac, “The quantum theory of the electron, Part 1,” Proc. R. Soc. (London) A, vol. 117, pp. 610-612, 1928.

[32] P. A. M. Dirac, “The quantum theory of the electron, Part II,” Proc. R. Soc. (London) A, vol. 118, pp. 351-361, 1928.

[33] Telegrapher’s Equation, Wikipedia, 2020. Available at: 27s_equations

[34] A. Giusti, “Dispersive Wave Solutions of the Klein-Gordon equation in Cosmology”, Scuola di Scienze Corso di Laurea in Fisica, University of Bologna, 2013, Available at: 31155754.pdf

[35] G. Evans, J. M. Blackledge and P. Yardley, “Analytic Solutions to Partial Differential Equations”, Springer, 1999. ISBN: 2540761241

[36] Integral Calculator. Available:

[37] Fourier Transform, Wikipedia, 2020. Available at:; Section 15: Tables of Important Fourier Transforms, Distributions; 15.4 One-Dimensional, Transformation 312.

[38] Green’s Function. Available:

[39] M. H. Silvis, “A Quaternion Formulation of the Dirac Equation”, Centre for Theoretical Physics, University of Groningen, 2010. Available at: