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Tetration and Hopf fibration?
#1
I have been watching these excellent (3) video lectures by E.Ghys in English online:

Ghys Video lectures Osculating Curves 2008

There is a lot of difficult and interesting stuff presented clearly. Some of that I do not understand.

Ghys speaks of osculating circles being disjoint. Of osculating conics having a subtler ordering than just being disjoint. Of Taylor polynomials being analogue to the disjoint osculating curves. Of differential equations representing these osculating curves, or, vice versa, these osculating curves being geometric representation of differential equations.

Then he proceeds to show Hopf fibration of a neighboroughood of a real point using crossing of 2 complex axis and sphere S3 around the crosssing. He arrives at knots etc. This is all fantastic.He says putting complex lines instead of real works like a local microscope which is farther enhanced by using higher and higher degree osculating curves =differential equations.

Now since Hopf fibration uses a pair of complex lines could it be that tetration allows to take the pair of points or values from INSIDE the local neighboroughood of a real point as revealed by Hopf fibration etc and compute the corresponding real value and vice versa?

Since Hopf fibration is topological, the values there are related to number of knots of ratios of areas, there are no coordinates inside there.

But please have a look at lectures, I may be misinterpreting something.Ghys also puts forward unsolved problems.

There is also an excellent film and web site about this stuff, in simpler way.

Dimensions Movie Ghys

Ivars
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#2
Ivars Wrote:Dimensions Movie Ghys

Ya thats a great movie. Though I dont see the connection to hyperoperations.
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#3
(04/09/2009, 12:55 PM)bo198214 Wrote: Ya thats a great movie. Though I dont see the connection to hyperoperations.

I am sure degree n of Hyperoperations are related to dimensions of complex projective spaces. I am not yet sure what is the role of Hopf fibration or Robinson Congruence or Penrose twistors in all this exactly, but I have no doubt the domain of tetration is complex projective space of dimension 4 or complex projective manifold of dimension 4 in CP5 or its (4 dimensional spherical null (at infinity) manifold) cut with a line in CP3 ( in line coordinates of CP5-Klein quadric)-which is essentially twistor space.

Same for pentation etc. CP5 or 5 manifold in CP6. etc.

The reason why I think so is the speed of operations. The higher up in complex projective space dimensions one goes, the faster calculations need to be peformed to project it on lower dimensional spaces. So, to make an instant change in CP3, and infinitely faster calculation has to be made in CP4 over all points or those representing a 4 manifold in higher dimensional projective space and then it has to be projected in into CP3. . And tetration is doing this (calculation) , as are other hyperfunctions. Such projection is akin to differentiation/integration

From here also arises in my opinion the strange coincidence between asymptotes of hyperoperations and values of Z function at negative even n which was present in one of the Andrews graphs. Riemann Z function also moves out of complex plane and across complex projective space dimensions, that is why it is so difficult to nail.

And also, the strange approximation for constant alpha from pentation fixed point value 1,85... I made a year ago or so.

This makes sense if one considers that cause of difference between EM and weak interactions could lie in CP5 or CP6. Such models of Universe exist. Only hyperoperations are missing.

Ivars
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