From Osher Doctorow Ph.D.
We are coming to a rather curious juncture in fractals and chaos research related to interdisciplinary growth processes. The good news is that we are making headway into interdisciplinary equations (logistic types especially), into dimensional analysis and its generalization Lie groups of point transformations and dimensionless ratios related to growth involving radiation (which expands from a source) and logistic and even exponential growth constants, and we are pushing our way into different phases including a more and more plausible superluminal phase. The bad news is that we still don't have a handle on growth in respect to expansion-contraction. One possible solution is to change our focus from expansion to contraction in an effort to understand similarity (e.g., self-similarity). Fractals have this tendency to create similar but smaller geometric objects on smaller and smaller scales, but it is difficult to find an equation form of this because of the difficulty of nth iterations or recursions or compositions. However, in geometry similar objects tend to have constant ratios at least in pairs of similar objects. The difficulty occurs in the fact that in a long sequence of similar objects, the ratios of sizes may changed for one constant to a different one and on and on. The Mandelbrot set give us a remarkable clue. It turns out that the sequence of disks as we proceed along the main direction of the Mandelbrot set (along the negative real axis from the origin) approaches a constant in their ratios of diameters, the Feigenbaum number, approximately 4.669 or 4.669201660910.... In mathematical language, with dn the diameter of the nthdisk (n = 1, 2, 3, ...), we have: 1) lim dn-1/dn = 4.669 approximately where the limit is as n approaches infinity (increases without bound). We can obtain a rate of contraction roughly analogous to the rates of expansion that we've been discussing logistically. This would be approximated by: 2) rate of contraction = lim (dn+1 - dn)/(dn+1) = lim (dn+1 - dn+1/4.7)/dn+1 = 1 - 1/4.7 where the Feigenbaum number 4.669 has been rounded off to approximately 4.7 Osher Doctorow Ph.D.