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Dynamic mathematics , tetration and fractal dimension of a "spiral"
#1
Hi,

I was reading Max Tegmark's "Mathematical Universe"

The Mathematical Universe
and he implies that mathematics are static, rigid structures.

I do not see it that way. Mathematics are dynamic and use Imaginary unit (and other imaginaries) as and imaginary time to order mathematical events.

For example, for me expansion in infinite series is a dynamic mathematical event, akin to explosion, which goes over scales like an avalanche, and, in case of non-dissipative mathematics, oscillate back to pull itself together.

For example, series
2I= I+ I/2+I/4+I/8+I/16 ......... can be seen as a critically pretwisted circle with imaginary radius I ( and circumference 2*pi*I - thus , imaginary) imaginary units selftwisting into smaller and smaller circles, making in the end "spiral" consisting of infinitesimal imaginary circles and then coming back to the original circle and than carrying the "momentum""to perform the same twisting in other direction, and so on. If math is not dissipative, it wil continue for ever. It would be a mathematical oscillator.

The partial sums of these series will then dynamically describe the sum of all imaginary radiuses in one half side of the initial ring.

Similarly I see Taylor expansion as it involves higher and higher integer derivatives, so each scale the mathematical avalanche passes through adds a some value.

Such series as above, giving sum of "lengths" can be viewed as fractal curve (Finite lenght infinite fractal "spiral") that is winding towards the center when series explode and adding length with each term, and unwinding from the center when series "gather together from infinite number of fractal pieces". And so it oscillates.

A finite partial sum can be viewed as similar process that for some reason has a smallest quantum- so it can not go further then certain term. The reason may be the engagement of mathematical object in some kind of mathematical structures that impose restrictions on its behaviour.

If we talk about infinite Products, that can be seen as FINITE Area/volume/hypervolume etc of n-dimensional ( and infinite in the end) object that is a combination of all fractal "lengths" of this imaginary "spiral".

The interesting thing about the partial Products is that in case of imaginary fractal "lengths" the Area (n=2) is negative, Volume (n=3) is - imaginary, hypervolume (n=4) is positive fraction, so this product kind of visits 4 different spaces in during each 4 cycle, but setlles in one space at infinite number of terms. And of course, in case of convergent series, this hyperfractalvolume is finite.

Now, there is one more type of series expansion which is not so popular- and that is infinite exponentials.

A= (ax)^(bx)^(cx)^(dx ) ........ this is a tricky one, since its not commutative nor associative, so the order of brackets is of importance, and leads to several different results.

Tetration ( right bracketing ) is one example.
The series with I as above will be then:
? = I^((((I/2)^((((I/4)^(((I/Cool^(((I/16)^((....)

What would be the geometrical meaning of this dynamic process in relation to winding/unwinding of fractal "spiral". It seems it has to to with n-th term mathematical dimension of such a spiral, which is obviously non-integer. In infinite case, that is the fractal dimension of the spiral as whole, winded, and its obviuous that this fractal dimension is somehow contained in the result which is currently (?). But since there is more then one way to produce such series via bracketing, I am not sure which to choose, yet tetration is what I intuitively prefer.

To sum up the above, the dynamic mathematical process of expansion in infinite series (products, exponentials) gives:
Sum= total fractal "lenght" of a winding "spiral"
Product=fractal "hypervolume" of the same "spiral"
Infinite exponentiation (right?Tetration?) = fractal dimension of the same "spiral"
For convergent series.


I am putting "lenght", "volume" in "" because I am not sure real number line is the only and the best ordering of numbers to use in these dynamic "spirals" . I think ratio curvature/torsion (or its inverse) of a line is a better represeņtation of numbers . In this case, the finite "lenght" of "spiral" would equal to Initial curvature/torsion ratio of a topological helix which gets unwinded in a set of discrete events- self twists, each next selftwist reducing/increasing that ratio by some value corresponding to term n.

The term spiral than get its meaning since such spiral can be constructed which has curvature/torsion ratio exactly as series have, after each turn ( as series have after each term).

Good thing about such "length" is that is does not take any space as it is intrinsic a property of a curve, and can be used in non-metrical spaces- e.g projective.

The fractal "hypervolume" in this case would be the multiplication of curvature/torsion ratios of each next twist .

And the fractal "dimension" would be the infinite iterated exponetatial of sequential curvature/torsion ratios after each twist.

One can easily see how such model of infinite series can be adopted to non-integer , continuos sequences- iterations - where also the curvature/torsion ratios in between full twists of initial helical circle are included.

For tetration series resulting in I its obvios that fractal dimension of such a process is Imaginary.

So if we make divergent series with divergent (fractal = all fractals are the same) "length":

e^(pi/2) + e^(Pi/2) + .......

With divergent (fractal) hypervolume :

e^(Pi/2)*e^(pi/2) *e^(pi/2) *..

We get Exactly 2 valued Imaginary fractal dimension- an undivadable pair:

h(e^(pi/2) = +- I .

Ivars
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