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Tetration and selfroot as a topological transformation of CP1 into itself
#2
I have just realized what I have said above:

SO important to make connection with ususal math since it helps when many things are everything is calculated already:


In tetration, we are dealing with Topological Reordering of Real number line (real number line is not quite like projective, since it has 2 infinities- + and - BUT : the infinity of CP1, Complex projective line is ALSO a double point- so spinor - so +- oo) to istself . Meaning there can be either finite, or infinite number of ways to transform it , reorder. Imaginary numbers , quaternions etc are needed to describe these reorderings.

Now, since INFINITY (in CP1) is a spinor +- oo, what is than e^pi/2?

Complex projective spaces have a circle S1 1-dim-sphere at infinity
So e^pi/2 is a parameter of that circle S1.

What parameters do circle has?

Area and Radius and circumference.

If area is e^pi, what would be circumference dimensionally? =e^+-pi/2. Radius?

Little tricky, but it is e. (or, still its e^pi/2 since than tetration would transform spinor "radius" to spinor "radius").

So: e^pi is an area of circle at infinity of CP1, e^+pi/2 -circumference, e - radius (or e^pi/2-radius, circumference I can not say right now- it may have no circumference at all).

What is +-Pi/2 then? The imaginary Dimension of CP1 (inside the line) . The initial twist of line .It seems, this TWIST is concentrated at point at infinity. It could be that the point at infinity is the most general idea we have abou the INSIDE of a line CP1.

So inside Real Number line, dimensions of space become imaginary, and is thus related to the usual ones we use.

CP1can be seen as a combination of 2 RP1 spaces with infinities +oo ( +Pi/2 imaginary dimension) and -oo ( -Pi/2 imaginary dimension) .


Also, I wanted to mention, that continuous topological deformation of real number line via tetration leads to states in between full automorphisms (full twist e.g I described ) that are fractal iterations of topology of real number line or CP1. These fractal states of real number line achievable via CONTINOUS deformation of it ( INCLUDING infinity!!!) have

NON-INTEGER DIMENSIONS of SPACE.


Best regards,

Ivars
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RE: Tetration and selfroot as a topological transformation of CP1 into itself - by Ivars - 12/04/2008, 08:15 AM

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