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Infinite tetration and superroot of infinitesimal
I would like to start with an possible physical interpretation of I as an imaginary dimension.

If You have a directed number on a line- vector AB in 1D, you can inverse it by turning it inside out via its origin- at first You contract it to 0 (point) while STORING continuously information about what it was - both size and direction- in imaginary dimension I. After that You release this information from imaginary dimension gradually until You get - AB on the same line.

To perform the same inversion by rotation, You have to create a picture of this imaginary dimension in plane with linear real number scale ( |AB*I|= |AB|) - and than we get complex plane, where the inversion is performed by rotation via this imaginary dimension I which we have to append perpendicularly to our real line at least for the time we perform the inversion which maintains all information required until we have rotated the initial vector by Pi or -Pi.

This implies that the process of inversion happens linearly ( so that as we reduce lenght of AB by dx the value in imaginary dimension grow as c*dx. This linearity leads to sin/cos relations in complex plane and also to exponetial dependance on imaginary angle I use below. Question- does it have to be- is it linear?

The question is I a unit of this imaginary dimension I or just the dimension with arbitrary units needed to perform "physical" actions like inversion of real space dimension via itself?

As an Example, one can take an infinitesimally thin shell elastic tube and inverse it by rolling it backwards on itself. During this rolling, the information about the length of the tube would be rolled in the very small torus it creates, while the information about direction would be retained in orientation of torus and relative Length of tube rolled in very thin torus and not rolled in yet or rolled out already. This would work equally to I as long as it works with infinitesimal shell perhaps of imaginary thickness,which I have to check in literature.

In this case this torus would play the role of imaginary dimension relative to our Real dimension we started with.

If, however, we would start with imaginary dimension and try to inverse it, we would have 2 imaginary dimensions, of which one would store information about the inversion of other.

The trick with quaternions seems to be that to keep such system closed in some sense, as Complex numbers which are algebraically closed- any algebraic equation in complex numbers leads only to complex roots- which helps greatly with series expansions - so that one imaginary dimension stores information about the inversion of other , while the other keep information about the inversion of former- seems kind of difficult because we get a double torus - but is it impossible?

in 3 Imaginary dimensions , we would have 3 toruses keeping information about each others inversions VIA ITSELF + also relative orientations. According to quaternions, they can not do it without the help of a scalar- or is it so? I have to try to picture this in more detail.

Next thing to ask is is there a natural metric on these imaginary toruses etc because it obviosly can not use real numbers to store this spatial information (it can if we ascribe real numbers to such dimension- like a+b*i+c*j+d*j, where a,b,c,d = real) - how does imaginary dimension store length since length is measured relative to another real line segment which is not given in imaginary space dimension.


Messages In This Thread
RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/11/2008, 09:27 AM

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