MANDELBROT ISO-SETS: ISO-UNIT IMPACT ASSESSMENT

In this introductory work, we use Santilli's iso-topic lifting as a cutting-edge platform to explore Mandelbrot's set. The objective is to upgrade Mandelbrot's complex quadratic polynomial with iso-multiplication and then computationally probe the effects on this revolutionary fractal. For this, we define the "iso-complex quadratic polynomial" and engage it to generate a locally iso-morphic array of "Mandelbrot iso-sets" by varying the iso-unit, where the connectedness property is topologically preserved in each case. The iso-unit broadens and strengthens the chaotic analysis, and authorizes an enhanced classification and demystification such complex systems because it equips us with an additional degree of freedom: the new Mandelbrot iso-set array is an improvement over the traditional Mandelbrot set because it is significantly more general. In total, the experimental results exemplify dynamic iso-spaces and indicate two modes of topological effects: scale-deformation and boundary-deformation. Ultimately, these new and preliminary developments spark further insight into the emerging realm of iso-fractals.

[1]  Konstantine P. Georgakakos,et al.  Estimating the Dimension of Weather and Climate Attractors: Important Issues about the Procedure and Interpretation , 1993 .

[2]  Ian D. Entwistle,et al.  Julia set art and fractals in the complex plane , 1989, Comput. Graph..

[3]  Christian Corda.  Introduction to Santilli iso-mathematics , 2013 .

[4]  Noisy nucleotides. DNA sequences show fractal correlations. , 1992, Scientific American.

[5]  D. Dingwell,et al.  “Explosive energy” during volcanic eruptions from fractal analysis of pyroclasts , 2006 .

[6]  Benoit B. Mandelbrot,et al.  Fractals and chaos : the Mandelbrot set and beyond : selecta volume C , 2004 .

[7]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[8]  Lloyd Demetrius,et al.  The origin of allometric scaling laws in biology. , 2006, Journal of theoretical biology.

[9]  James H. Brown,et al.  The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization , 2005, Journal of Experimental Biology.

[10]  Yuval Fisher.  Exploring the Mandelbrot set , 1988 .

[11]  Mitsuhiro Shishikura.  The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets , 1991 .

[12]  Massimo Negrini,et al.  Genetic chaos and antichaos in human cancers. , 2003, Medical hypotheses.

[13]  Benoit B. Mandelbrot,et al.  Fractals and an art for the sake of science , 1989, SIGGRAPH '89.

[14]  Nathan O. Schmidt.  Dynamic Iso-Topic Lifting with Application to Fibonacci's Sequence and Mandelbrot's Set , 2013 .

[15]  John Briggs,et al.  Fractals : the patterns of chaos : a new aesthetic of art, science, and nature , 1992 .

[16]  S. Grant,et al.  Scaling theory and solutions for the steady-state coagulation and settling of fractal aggregates in aquatic systems , 1996 .

[17]  P Philippe,et al.  Chaos, population biology, and epidemiology: some research implications. , 1993, Human biology.

[18]  D R Fish,et al.  Three-dimensional fractal analysis of the white matter surface from magnetic resonance images of the human brain. , 1996, Cerebral cortex.

[19]  J. Elsner,et al.  Chaos, Strange Attractors, and Weather. , 1989 .

[20]  P. Duesberg,et al.  Chromosomal chaos and cancer. , 2007, Scientific American.

[21]  Xu Nan,et al.  The fractal dimension of EEG as a physical measure of conscious human brain activities , 1988 .

[22]  Christian Corda.  Introduction to Santilli iso-numbers , 2012 .

[23]  Edgar E. Peters.  Fractal Market Analysis: Applying Chaos Theory to Investment and Economics , 1994 .

[24]  Blake LeBaron,et al.  Chaos and nonlinear forecastability in economics and finance , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[25]  Reza Katebi,et al.  Initiating Santilli's Iso-Mathematics to Triplex Numbers, Fractals, and Inopin's Holographic Ring: Preliminary Assessment and New Lemmas , 2013 .

[26]  James H. Brown,et al.  The fourth dimension of life: fractal geometry and allometric scaling of organisms. , 1999, Science.

[27]  Jaime Sanudo,et al.  Fractal dimension of lightning discharge , 1995 .

[28]  Stephen L. Harris,et al.  Agents of Chaos: Earthquakes, Volcanoes, and Other Natural Disasters , 1990 .

[29]  J. Valdivia,et al.  Model of gamma ray flashes due to fractal lightning , 1999 .

[30]  Gary William Flake,et al.  The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation , 1998 .

[31]  James H. Brown,et al.  A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science.

[32]  Robert F. Cahalan,et al.  Fractal Statistics of Cloud Fields , 1989 .

[33]  E. Bullmore,et al.  Adaptive reconfiguration of fractal small-world human brain functional networks , 2006, Proceedings of the National Academy of Sciences.

[34]  P. Krehbiel,et al.  Three‐dimensional fractal modeling of intracloud lightning discharge in a New Mexico thunderstorm and comparison with lightning mapping observations , 2007 .

[35]  C. E. Puente,et al.  The Essence of Chaos , 1995 .

[36]  J. Rogers.  Chaos , 1876 .

[37]  J. Huisman,et al.  Biodiversity of plankton by species oscillations and chaos , 1999, Nature.

[38]  Sammis,et al.  Fractal distribution of earthquake hypocenters and its relation to fault patterns and percolation. , 1993, Physical review letters.

[39]  강태원,et al.  [서평]「Chaos and Fractals : New Frontiers of Science」 , 1998 .

[40]  A. Brunt,et al.  Plant Viruses, Unique and Intriguing Pathogens – A Textbook of Plant Virology , 2000 .

[41]  G. Casella,et al.  A statistical model for the genetic origin of allometric scaling laws in biology. , 2002, Journal of theoretical biology.

[42]  J. Tennyson.  In the wake of chaos. Unpredictable order in dynamical systems , 1995 .

[43]  Xu,et al.  Periodicity and chaos in a one-dimensional dynamical model of earthquakes. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  C. Scholz.  Earthquakes as chaos , 1990, Nature.

[45]  G. Gilbert.  Book Review of The computational beauty of nature: Computer explorations of fractals, chaos, complex systems and adaptation. Gary William Flake , 2000 .

[46]  J. J. Ventrella,et al.  Evolving the Mandelbrot Set to Imitate Figurative Art , 2008 .