From Osher Doctorow Ph.D.
Jerry Iuliano, whose email is JerryIuliano@aol.com, posted some very interesting comments to me today on the subject Omega-2 constant and TOE. Jerry is far more familiar with number theory and numerical constants than I am, and he is interested in (and has been pursuing for a long time) both a unified theory of fundamental forces in physics and Fermat's Last Theorem. At a number of points in my theorizing when I have touched on matters related to numerical constants of considerable importance, Jerry has emailed me with extremely valuable ideas, some of which I have pursued and some of which I wished that I had. As a Non-Mainstreamer myself for a long time, I can say with little doubt that Jerry is even more Non-Mainstream than I am in his particular fields - he isn't embarrassed at pointing out Ancient Egyptian and Biblical references to or occurrences of various number constants, while I tend to look over my shoulder in embarrassment at those points at least when discussing them on scientific-mathematical forums. Jerry's posting concerned the Omega-2 and Omega-One Constants and Transformations, but he mentioned their tie-in with Weierstrass Elliptic Functions which are involved in Fermat's Last Theorem's proof and which in turn tie in with Fibonacci numbers which are of considerable interest in plant growth and other contexts. I looked up Jerry's references to Wolfram and Eric Weisstein's World of Mathematics, and reviewed some characteristics of the Weierstrass Elliptic Function that now take on a new significance for me at least. Letting r(z) be the Weierstrass Elliptic Function in my terminology, one of the most remarkable equations is: 1) r'^2(z) = 4r^3(z) - g2r(z) - g3 where g2 and g3 are elliptic invariants which, in the equianharmonic case, are respectively 0 and 1. Readers will recognize the growth equations in (1) except that with r' as dr/dz, r'^2(z) is used instead of r'(z) as in typical growth equations and the cube of r is one degree higher than in most growth equations for the relevant function taking the place of r. The Weierstrass elliptic curve modular form is equivalent to the "golden number" of ellptic modular forms, this number being the Omega-One constant which is (1/2)w2(1 + isqrt(3)) where w2 is called the Omega-Two constant defined by the value 1.529954037... according to Jerry. If we follow up the Golden Mean or Golden Ratio Phi, for example in the Eric Weisstein source mentioned, it turns out to be the limit lim Fn/Fn-1 of successive Fibonacci numbers as n --> infinity, but the latter ratio Fn/Fn-1 also measures fractions of turns between succesive leaves on stalks of various plants in botany. Fn itself turns out to be of considerable interest in biological growth - for example, it is the number of pairs of rabbits n months after a single pair begins breeding if newly born bunnies are assumed to start breeding when 2 months old. Phi turns out to be the most irrational number because of its continued fraction representation entirely by ones. Osher Doctorow Ph.D.