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Infinite tetration and superroot of infinitesimal
I was looking at these graphs:

If You plot in polar coordinates (r,phi) following spirals :

r= phi^(1/phi) , r= phi^(1/phi)^phi, r= phi^(1/phi)^phi^(1/phi) etc they all seem to converge to a circle with a radius e^(1/e) so that is just its area. But I have no software to check it, so it might be a mistake. May be it is smaller and has no connection to the problem.

Anyway, it is a very strange graph crossing itself once at angle which seems not to be 5/2* pi - after one full rotation - but again, I have no software to check. With each rotation, the angle of crossing gets smaller, there also seems to be a limit angle of crossing, which seems bigger than pi/4 but that is after 5000 rotations -I can not simulate more.

Now we think that infinity^(1/infinity) which was the beginning of this thread is not very well defined. Looking at the polar graph of r=phi^1/phi it does not seem so. First, there is well defined limit circle, I suppose, which has area.

Second, there is a limit finite angle where outgoing spiral at phi< pi/2 cuts the inner limit circle.

Third, there is the lenght of the spiral from origin ( which is also undefined, 0^1/0) to the point where it crosses limit cycle.

So , in fact, phi^1/phi etc. defines at least 3 values, 2 of them 1 dimensional, which could be used as limits for any of limits we are looking for:

e.g. infinitesimal^1/infinitesimal = lenght of the arc till crossing
infinity ^1/infinity = angle of line from center to the crossing point, or, alternatively, tangent angle of the spiral at that point- in fact, there are 2 tangent angles-another to the circle at that point. Plus area of circle.

if we change sign of r=-phi^(1/phi), spiral wounds the same way but with -pi phase offset.
If we change also sign of exponent, r=-phi^(-1/phi) things spiral appears from infinity to fill the cycle from inside. in this case, there is no arc length, but 2 tangent angles + area is still there. The same with r=phi^(-1/phi), offset by pi.

In both cases with negative exponents spirals turn anticlockwise , also in both cases with positive exponents. How ever, if we consider the limit cycle (ring) as the origin of spirals, spirals they wind in opposite directions, kind of squeezing the limit cycle in the middle.

So why not define the limits using the finite tangent angles? Or their trigonometric functions, e.g tangens? The one of the outgoing/incoming spiral for infinitesimal^(1/infinitesimal), the one of the limit circle at crossing for infinity^1/infinity.

There are options to define those limits , why no one is used?
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Messages In This Thread
RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/06/2008, 07:43 PM

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