26 He listed situations where, in contrast to the classical paradigm, incidents do not compensate for each other, but are additive, #Protein Tyrosine Kinase inhibitor randurls[1|1|,|CHEM1|]# and where statistical predictions become invalid. He described his theory in a book,27 where he presented what is now known as the Mandelbrot set. This is a fractal defined as the set of points c from the complex Inhibitors,research,lifescience,medical plane for which the recurring
series defined by zn+1 = zn 2 + c, with the condition z0 = 0, remains bounded (Figure 3). Figure 3. The Mandelbrot set a point c is colored black if it belongs to the set and white if not. A characteristic of fractals is the repetition of similar forms at different levels of observation (theoretically at all levels of observation). Thus, a part of a cloud looks like the complete cloud, or a rock looks like Inhibitors,research,lifescience,medical a mountain. Fractal forms in living species are for example, a cauliflower or the bronchial tree, where the parts are the image of the whole. A simple mathematical example of a fractal is the so-called Koch curve, or Koch snowflake.28 Starting with a segment of a straight line, one substitutes the two sides of an equilateral triangle to the central third of the line. This is then repeated for each of the smaller segments obtained. At each substitution, the total length of the figure increased
Inhibitors,research,lifescience,medical by 4/3, and within 90 substitutions, from a 1 -meter segment, one obtains the distance from the earth to the sun (Figure 4). Figure 4. The first four interations of the Koch snowflake. Fractal objects have the following fundamental property: the finite (in the case of the Koch snowflake, a portion Inhibitors,research,lifescience,medical of the surface) can be associated with the infinite (the length of the line). A second fundamental property of fractal objects, clearly found in snowflakes, is that of self similarity, meaning that parts are identical to the whole, at each scaling step. A few years later, Mandelbrot discovered fractal geometry and found that Lorenz’s attractor was a fractal figure, as are the majority of strange attractors. He defined fractal dimension (Table I). Mandelbrot quotes, Inhibitors,research,lifescience,medical as illustration of this new sort of randomness, the French coast
of Brittany; its length depends on the scale at which it is measured, and has a fractal dimension between 1 and 2. This coast is neither MTMR9 a one-dimensional nor a two-dimensional object. For comparison the dimension of Koch snowflake is 1.26, that of Lorenz’s attractor is around 2.06, and that of the bifurcations of Feigenbaum is around 0.45. Thorn, Prigogine, and determinism again René Thorn is the author of catastrophe theory.29 This theory is akin to chaos theory, but it was constructed from the study of singularities, ie, continuous actions that produce discontinuous results. Catastrophe theory is interesting in that it places much emphasis on explanation rather than measurement. Thom was at the origin of a renewed debate on the issue of determinism.