Hi IVARS!
A small preliminary comment (during the weekend I shall supply more) to your interesting "vision":
Actually, three dimensions are already there:
- y is the dimensionn [h(b)] where the tetration "heights" are measured ("vertical" in the traditional, so to say, representation);
- b is the dimension allowing us to define the "bases" (horizontal, in the same convention);
- x is the dimension perpendicular to y and b (a cross section perpendicular of the yb plane, perpendicular to the worksheet, in the same conventional representation).
On the yx cross sction planes (allow me to say so) the plots of the tetrational functions are, indeed (in my opinion, of course) oscillating lines plotted in a 2-dimensional plane, limited by maxima of y defined by the even integer values of x (0,2,4,6,...) and minima of y defined by their odd (1,3,5,...) integer values. These plots show, in the available approximated simulations, a decreasing, but not vanishing, oscillating behaviour in the region 0 < b < e^(-e). In fact, their asymptotical shapes, for x -> + oo, seem to be describable as "permanent oscillations" around an asypmptotic axis, defined by:
y = plog(-log/b x)(-log/b x).
Therefore, I agree with you (IVARS) thet we need an extra dimension, but I think that it should be represented by the imaginary axis itself. In this case, it will be the "fourth dimension" of the diagram.
I also agree with your "spirally-like vision", but I guess that the (deformed) spirals (for any fixed b) should be ... unique. What we "see" in the yx plane is (I guess) the "real projection" of the spiral, at any fixed b. On the contrary, what we see in the "traditional" yb diagram (the upper and lower h's) is (always in my guessing dream) the projection of the spiral on the yb plane, for x -> *oo. The upper and lower y, in the "Forbidden Zone for Integer Superexponents" should show the upper and lower maximum elongations around the center of the spiral, for any constamt b and for x -> +oo.
Sorry for the ... approximated description and thank you for your kind attention.
GFR
A small preliminary comment (during the weekend I shall supply more) to your interesting "vision":
Ivars Wrote:.............
When I was speaking in previous post about the missing dimension, I was thinking about rotating plane (or rotation in plane) perpendicular the graphs of GFR and parallel to y axis.
Then I have not even a guess, but just a vision of these 3 definitions of h(x) forming a double helix (odd/even) with a thread (x^1/x) in the midlle in region e^-e<x<e^(1/e). The projection on GFR graph is just a single line, but projection on that perpendicular plane is 3 points at each x.
When bifurcation happens x< e^-e, we get 2 opening spirals rotating left (odd?) and right (even?) in that plane and I am not sure what happens with x^(1/x) central thread there-it may tend to become a rotating connection between these spirals of that circle.
Ivars
Actually, three dimensions are already there:
- y is the dimensionn [h(b)] where the tetration "heights" are measured ("vertical" in the traditional, so to say, representation);
- b is the dimension allowing us to define the "bases" (horizontal, in the same convention);
- x is the dimension perpendicular to y and b (a cross section perpendicular of the yb plane, perpendicular to the worksheet, in the same conventional representation).
On the yx cross sction planes (allow me to say so) the plots of the tetrational functions are, indeed (in my opinion, of course) oscillating lines plotted in a 2-dimensional plane, limited by maxima of y defined by the even integer values of x (0,2,4,6,...) and minima of y defined by their odd (1,3,5,...) integer values. These plots show, in the available approximated simulations, a decreasing, but not vanishing, oscillating behaviour in the region 0 < b < e^(-e). In fact, their asymptotical shapes, for x -> + oo, seem to be describable as "permanent oscillations" around an asypmptotic axis, defined by:
y = plog(-log/b x)(-log/b x).
Therefore, I agree with you (IVARS) thet we need an extra dimension, but I think that it should be represented by the imaginary axis itself. In this case, it will be the "fourth dimension" of the diagram.
I also agree with your "spirally-like vision", but I guess that the (deformed) spirals (for any fixed b) should be ... unique. What we "see" in the yx plane is (I guess) the "real projection" of the spiral, at any fixed b. On the contrary, what we see in the "traditional" yb diagram (the upper and lower h's) is (always in my guessing dream) the projection of the spiral on the yb plane, for x -> *oo. The upper and lower y, in the "Forbidden Zone for Integer Superexponents" should show the upper and lower maximum elongations around the center of the spiral, for any constamt b and for x -> +oo.
Sorry for the ... approximated description and thank you for your kind attention.
GFR