From Osher Doctorow Ph.D. mdoctorow@comcast.net
I posted yesterday to geometry-research in the Math Forum (which can be accessed by keywords Math Forum or geometry-research or geometry- research@forum.swarthmore.edu) the posting "Geometric Existence" which basically strongly indicates that for growth-expansion-contraction processes and events, including optimally controlled processes and events, Geometric Existence is the Inverse or Opposite (in a certain sense) of Geometric Change. This explains the exponential bacteria growth and radioactive decay equations, the logistic population growth equation in a limited-supply environment, and the Riccati optimal controlled growth equations, and much more. For those who know Calculus (not required to roughly understand these things) or Differential Equations (similarly), Geometric Change is the Derivative, while Geometric Existence is the Integral, under growth scenarios. For those who don't know Calculus, it turns out that lengths, areas, and volumes are direct expressions of Geometric Existence under growth situations. Hence, the area of the Mandelbrot Set becomes of critical importance. Robert P. Munafo, 2000 Aug 31 in "Center of Gravity, Mu-Ency at MROB," www.mrob.com/pub/muency/centerofgravity.html, also stated to be from Munafo's Mandelbrot Set Glossary and Encyclopedia, 1987-2003, points out that the Center of Gravity of the Mandelbrot Set is obtained similarly to calculating the area of the Mandelbrot Set, and is very closely approximated by: 1) (ln(3) - 1/3)^Feig1 where Feig1 is the first Feigenbaum constant, equal to .28676 82633 82935 02685 29586... However, the first Feigenbaum constant is the (limiting) ratio between the areas of each circular region (main regions in a sense) and the next smaller one in moving to the left from the origin in the Mandelbrot set. Although it may be premature of conclude that we understand this, it seems as though Geometric Existence is central to Chaos and thereby to Fractals. I would guess that the Universe in its formation was on the borderline of Chaos if not in Chaos, and that this was an expression of the nature of its early Geometric Existence and also Geometric Change (expansion, etc.). Osher Doctorow Ph.D.