Hi Ray,
You write, "If the Golden Ratio is important to Nature, then how should it express itself?"
Actually, I don't think it _is_ important to nature, i.e., as a physically real element. The important properties of the ratio--proportion and recursion, as evidenced in the Fibonacci sequence--are shared with the line of positive real integers, R_+. As sufficiently large ratios of Fib_m/Fib_n converge to the limit 1.618 ..., sufficently large ratios of n/n+1 converge to unity. Which convergence do you think is more important?
The properties of an uncompressed arithmetic sequence are far more powerful analytically. Take Euler's equation, E^i*pi = -1. It tells us how to find the origin of the complex plane, so as to identify those fractions of unity on which every real function lives. It gives us a "lever," in a colloquial manner of speaking, by which changes in relations among points--whether mathematical or physical--can be calculated in up to four dimensions, and extended analytically to n-dimension or infinite dimension domains.
Compare this continuous function analysis, incorporating a complete algebra, with the numerology of discrete objects to which one assigns meaning by fiat. There are important technical reasons why field theories reign in physics, which have nothing to do with the intransigence of old-fashioned thinkers; clearly, the field of 2 dimensions is the fundamental playing surface on which we measure and experience.
That nature exhibits patterns of the Fibonacci sequence (leaf growth, sunflower seed arrangements, etc.) does not of itself imply that the patterns are discretely encoded in fundamental physics. Rather, we might interpret these artifacts as discrete 1-dimensional views of n-dimension origin--which makes far more sense to me.
You wrote,
"You (Tom) said 'A scale invariant universe admits infinite self similarity of quantum elements'. What if infinity becomes a 'stable infinity'? Would we now have a finite number of self similarity of quantum elements? My latest model (which is diverging radically from E-Infinity) seems to have a finite Multiverse. Please read Len Malinowski's ideas at http://www.scalativity.com/
I'll visit the site. I don't know what one means by "stable infinity," however. Infinity is "stable" by definition; i.e., dynamics cease. We can't do analysis on infinity; it's not a number.
By my comment that you cited above, I mean that scale invariance does not obviate multiscale variety (Bar-Yam, et al), such that systems and subsystems that evolve and cooperate at different temporal rates in 4-dimension spacetime can be modeled discretely in an infinite dimension (Hilbert space). If you read my paper, you will see that I suggest continuation of R^n with H^infinity. That's why I like to see if Iovane is on the same track with the paper I mentioned before. I just don't know what "Cantorian spacetime" is, or why one would deem it either physical or theoretically necessary.
Tom
