The Complementary set of Anti M-Set Fractals
I. The Universal Pair Principle, Complex numbers and Fractals
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"And Noah and his sons and his wife and his sons' wives entered the ark to escape the waters of the flood. Pairs of clean and unclean animals, of birds and of all creatures that move along the ground, male and female, came to Noah and entered the ark, as God had commanded Noah." (Genesis, 7,7) Pairs are a fundamental principle of nature, both of still physics and life. The pairs could be Complimentary to each other or Antagonistic, and many a time they are both at the same time. There are pairs everywhere and at each level of nature and existance: from the obvious visible male and female on to the living body with its pairs of almost all organs: pairs of hands, legs, eyes, ears, etc., then going on to the inner organs like lungs and kidneys, and to smaller organelles like the adrenal glands; continuing down to the cellular and macro-molecular level, where antigens meet their specific antibodies antagonists, and to the fundamental basis of all, where the double helix chains of the DNA are constructed with 2 pairs of antagonistic-complimentary building stones. The Antagonistic-Complimentary Universal Pair Principle is carried as deep as the elementary structures of the molecular bonds and atomic particles: electrons and positrons, up quarks and u-bars, and so on. Even single organs of the living body (human for example) are anatomically a fusion of two organs: the heart with its left and right chambers and ventricles held together by a firm middle septum, and the brain who has obvious right and left parts, being connected together only by a specific connecting organelle - the "corpus callosum". The concept that fractals could be regarded as models for the unpredictable morphology, anatomy and events of nature has originated by Benoit Mandelbrot himself: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." and further developed in his speech "The Fractal Geometry Of Nature", San Francisco 1982. Combining the fractal geometry of nature theory with the universal pair principle I started the search for the Anti Fractals with the help of my own fractal generator DigiGraph 2003). All fractal pictures that are presented in these pages were generated with DigiGrah version 5.0. Fractals are dealing with Complex numbers. By definition a complex number is a quantity that has two elements: a + ib. In this format a and b are Real numbers and the symbol i stands for the Imaginary numbers unit. By definition imaginary number is the positive square root of -1. The C set of the complex numbers is well presented on a 2D plane by using the the x,y coordinates. Thus complex numbers are Pairs and two-dimensional while real numbersare only one-dimensional. This is the connecting point between the Universal Pair Principle and the fractals math. The X axis of the 2D plane represents the real numbers and the Y axis the imaginary numbers, and the complex fractal pixels are plotted on this very plane. Evidently the complex plane on which the fractal is plotted is enough to establish the pair principle of the fractalgrams. Nevertheless I was still looking for a definite Anti Fractal fractalgram, equivalent to the M-Set fractalgram. Of course this should be looked for at the formula level, and the Anti Fractal should be generated with the same methods that generated the fractal itself. Author: Dr. Joseph Trotsky |
