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Infinite tetration and superroot of infinitesimal
Well I hope Henryk will not punish me for these wild conjectures. This is the place I can store them and hope to get some feedback.

I studied what i could for 2 weeks and understood little, got lost almost totally but also understood that most books on whatever start with the same- real vector spaces , and then go on developing the ideas. Most who write today about quaternions or not seem not to have studied Hamiltons Elements of quaternions since he gives there much more than just vectors to be used for 3D rotations in visualisation. He develops quaternion mathematics- functions, series, calculus, links to geometry, etc. It is not mentioned anywhere in new books even about quaternions. Even first explanator of quaternions Tait did not use most of that. But that is another story to be looked at.

Briefly, if we compare number types geometrically with straight line elements:

Scalars are lentgh of line, or ordinary numbers. They also may measure the translational motion of one end into other.

Vectors are directed ordinary numbers, obtaine by inverse summation : A-B . e.g 7-5 is a directed number 2 and is a vector. While -2 is the same vector in opposite direction. Normal imaginary unit have been related to these a Amplitude*phase.

Than , by same logic we must (?) have numbers that are obtained by inverse multiplication, i.e. division. It seems quaternions are these, as Grassmann or Clifford defined his algebra from existance multiplicative inverse, of which there are only 4 known. For some reason, it involves imaginary space of 3D and is related to rotations in space. What is rotated in imaginary space around origin is ordinary imaginary number I- point on imaginary 2-sphere- which is the same as I for normal complex numbers, but generally not the same as pure quaternion axis i,j,k - none of them. x^2+1=0 has infinitely many solutions- +- I -is -antipodal points on this imaginary 2-sphere in pure quaternion space.

Then, next we may have numbers obtained by inverse exponentiation, or log. They may be twisted (?) numbers. Or what else could we do to an imaginary line segment (I have not figured out how and why) . e.g. Because octonions are non-associative, as is not exponetiation usually ( x^y is not y^x usually). Twisted numbers may oscillate?

Then, we move to 4th operation, or tetration. What numbers we may obtain by inverse tetration? The inversion of infinite tetration is self root. Self root may denote looping numbers. But for self root to be inverse of tetration, we need infinite tetrations. In the middle, as long as number of tetrations is finite, we can get inverse by various order superrots as long as we tetrate one and the same variable ( number) . We may need another set of numbers or we may be able to survive with combinations of existing. Basically what we can add to number as imaginary line after translating, directing, rotating, twisting it is what? It may be further internal twists of the imaginary line? if that is related to sedenions, what could it mean geometrically and why it is not anymore divisor algebra?

The reason I mention this in tetration forum is that it seems to me that numbers have internal structure, which may well be imaginary since we have no idea about it when we look at number line. It only reveals itself under proper operations. So number line looks (perhaps) as a rope which has internal degrees of freedom, brought out under fast enough operations applicated enough or proper amount of times. This imaginary structure at least at integer operation numbers may be related to imaginary numbers and their geometric developments into quaternions, octonions, sedenions etc. ...


Messages In This Thread
RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/10/2008, 12:26 PM

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