01/22/2008, 09:46 PM
GFR Wrote:@ IVARS
Please see the attachment, in which I try to imagine the situation represented in a 3-dimensional environment (y,b,x). The plots of y=b#x are the (appromimated and simulated) real projectionns on the yx plane of the y=d#x complex helicoidal "functions". The transition (yellow, off-limit) zone on the yb diagram is (in my ... simple opinion) the graph of the max-min span of the undetermined y values, when x -> oo. Persintent oscillations. Difficult to describe, ... see the plots.
GFR
I was studying it so long but its still above my head. I wanted to see oscillating behaviour as n=infinity (both odd and even) in region x<e^-e- so that the jumps between 2 infinite tetration of x where one is odd, one is even, next odd again etc. happen via rotation in perpendicular plane. As both h odd and h even as n-> infinity are continuous functions, if we infinitely backtetrate them , there has to be a distance on x axis between the results, but its infinitesimal.
So the spiral ,or circle in perpendicular plane will consist of infinitely close planes , but nevertheless will have distinctly separate endpoints in plane xy as points of continuos functions h odd (x) h even x .
So in real plane xy we do not see any difference between x for 2 neighbouring values of infinite tetration 2n-> infinity and 2n-1 -> infinity, but we know there must be an infinitesimal difference so perhaps it is visible in perpendicular space as a direction of rotation of each of these spirals - i can not imagine any other way left to differentiate them.
I guess i was not clear enough, but do not have the skills to make or post a picture. In the end , it should be possible to explain such things in plain words though it might take a lot of space
