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Infinite tetration and superroot of infinitesimal
#51
To make it shorter:

h(i^(1/i)) = 1/i
h((1/i)^i) = i

-ith root of -i = selfroot of -i = reverse to infinite tetration of ((1/i)^(i)) = e^(pi/2)
ith root of i = selfroot of i = reverse to infinite tetration of ((i^(1/i))= e^pi/2

Self roots of both i and - i are equal and give 3 new values in one given combination: e, pi and 1/2 plus 4th value 4,81......... .
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#52
I have a little problem now:

i^(1/i) = e^pi/2

(1/i)^i = e^pi/2

but we know geometrically,in complex plane to compute i^-i we take e^(ipi/2)*(-i) and to compute -i^i we take e^(-ipi/2)*i

The exponents ipi/2 and -ipi/2 are complex numbers as such. So multiplying them by i or - i means rotating them in opposite directions.

If we take i*(pi/2) and multiply by -i, we rotate it to the right, clockwise
If we take -i*( pi/2) and multiply it by i, we rotate it to the left, anticlockwise.

in both cases we end up with pi/2 but where did we lose the information about the direction of rotation we had to apply taking selfroots of i and -i ? The directions were opposite, but we end up with 1 number. I do not like it at all, this loss of information.

Now we know that taking self root of i, or -i is like backtetrating from i or -i to e^pi/2. This must happen in some steps via some space, like infinite number series or limits travel through real number space .

Why we do not see that space where these opposite turns of i and -i are happening with every backtetration? Do we miss some dimension? Like spin dimension in physics?

Ivars
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#53
A relevant question to previous:

If we take a differential of Imaginary unit, dI ,which is imaginary infinitesimal (also mentioned by prof. Bell as i*epsilon):

1) What would be its orientation relative to I ? ( most likely 90 degrees )
2) In which space we would see this orientation?
3) If multiplying by I means rotation by 90 degress in complex plane, would multiplying by dI mean infinitesimal rotation in what plane?
4) What is self root of dI? Or what tetration would result in dI?

Ivars
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#54
Ivars Wrote:If we take a differential of Imaginary unit, dI ,which is imaginary infinitesimal (also mentioned by prof. Bell as i*epsilon):

Ops ... sorry! I didn't get you.
In mathematical analysis, by "imaginary unit" we normally mean a constant, given by the famous sqrt(-1), which actually gives {-i,+i}.

It can also be defined as the solution of equation: -x = 1/x. No real number can satisfy it and therefore, this new mathematical object is around since the 18-th century. It is a constant and, therefore, I presume that we must have d(i) = 0. Take a complex number:
z = x + iy, well then: dz = dx + i dy.

Other strange objects are around, like quaternions, octonions, senonions (I presume so!), with multiple different imaginary units and with complicated inter-relations, properties and symmetries. But, also there, all these different "units" are constant.

The only "imaginary differential" that I can figure is someting like i dx or alike. And, in this case, i dx is always "parallel", so to say, to i. Perhaps I am missing the point.

Gianfranco
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#55
Thanks, GFR,

I know that , of course. I am deliberately trying to find a true imaginary infinitesimal, not i *epsilon, sorry, not i *dx.

Well if we begin with i as infinitely rotated into higher dymensions length (e^pi/2) so that -i = h(e^pi/2) is a hypervolume of an infinite hyperdimension. OK? Totally unclear , no physical content, but that is what it could be.

If we than remove one lenghts e^pi/2 , back tetrate, so to say, what will happen with that hypervolume? Will it stay the same? or will it be one hyperdimension smaller?

Would it than be the same -i or, perhaps, -i - di? So that difference between 2 such hyperdimensions is di, meaning that to get -i, You have to integrate over all hyperdimensions there is of di-> which equals infinite tetration of e^(pi/2).

I know i is considered to be constant, but I do not see why that has to be accepted. i, pi, e are just symbols, and their content is in fact, infinite of unknown since they have no physical analoques.

di = 0 is an axiomatic assumption that works well in math. The same for d(pi)=0. Fine. And than we have Godels law and happily watch exponential growth and specialization of mathematics because we have limited ourselves to the usage of well defined symbols and application of well defined operations.The number of finite symbols and operations will grow infinitely. The unification theories can not be based on such approach as by definition the deeper You go, the bigger diversity You will get. That is just obvious, so following the same logic and hoping that suddenly string theory will unite everything is just impossible- the complexity of picture will just increase. Today we have 10^500 string theories that might fit- than we will develop even deeper understanding- and there will be 10^500^500 superstring or whatever theories appearing. There is no end to this, and Godel's theorem say exactly that. The same applies to mathematics as such - the more You apply the principles and axioms, the more diverse and incoherent picture You will get although there will be visible similarities in many branches, the proofs that these similarities are rigorous will grow exponentially bigger and more difficult to find.

But in fact, using ill defined or undefined symbols like infinity and applying logical rules to them
we can go around Godels theorem and build a logically coherent system with undefined or not so well defined inputs-which is a small price to pay.

e.g. I is imaginary unit and equals infinite hyperspace volume. dI is a change of dimension of this hyperspace by one while still close to infinity. Very ill defined-no understandable content as infinity is involved.

Then we apply the rules of differentiation/integration or may be invent new ones and see what happens. i appears in so many mathematical formulas that it will not take long before some pattern might emerge or might not. If not, we may change the rules, or intial bad defintions to even worse, and start again.

It should not take long before the right rules can be found, or, their inexistance proved. If inexistance is unprovable because of the same Godel's theorem- than it is possible.






Ivars
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#56
Hi, Ivars!

Mmmmm ..., as Gottfried says. Let me think about that.

GFR

NB.: "Liguistically" modified on 2008-01-24 at 21h38. GFR
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#57
One way to find dI would be from lnI/I=pi/2.Assuming pi/2 constant for time being, we get:

d(lnI/I)=0

dI/(I^3)-lnI/I^2=0

dI = IlnI = -(lnI)/I = ln (I^I)=-ln(I^(1/I)= -pi/2 so that if

-I= h(I^(1/I)), dI = -ln(I^(1/I)).

so we can have
-IdI = -W(- ln(I^(1/I)) = I*(Pi/2)
IdI= W(-ln(I^(1/I))= -I*(pi/2)

We can integrate dI , IdI , etc over all hyperdimensions or finite interval of and get some values. First impression is that pi is not a constant moving from one hyperdimension to next, otherwise we get non-identities, but this have to be little experimented with.

And so on. More later.
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#58
Without bothering about pi and signs, in principle we can define lines and surfaces in hyperdimensional space as

Integral (a to b) di is a line, integral (1-infinity ) di is = - i is also a line but infinite.
We also have integral (-infinity to 0?) = i which is also some infinite line. Then Integral (-infnity to plus infinity) = -I + I = 0 Is a linear combination of those 2 lines- the lines itself are most likely undefined spirals of opposing rotation, but perhaps even not so well specifiable in general case.

Probably we can construct "area " of "surface" in hyperdimensional space as
Integral over all surface of dI*dI where dI*dI is a surface element in hyperdimensional space
and so on-volume, 4 volume .......... infinite dimensional volume.

while pi/2 is assumed constant, "area " element dI*dI = -pi/2*-pi/2 = (pi^2)/4


and "volume" element dI*dI*dI = -(pi^3)/8.

If dI is imaginary infinitesimal, it is possible to create (1/dI) = imaginary infinity

1/dI = I/lnI= (-2/pi). So far so good.

In fact, it seems pi is not constant as hyperdimensions shift by dI. So, as long as 1/2 is constant,

dI=d(pi/2) = -1/2d(pi)

With this, element of area is 1/4 d(pi)^2. Interestigly, that relates to black hole area rather nicely, just need to take integral over d(pi) over involved hyperdimensions.

if dpi is a change in pi, than Intergral (-1/2)dpi) = - I = e^(-I*pi/2)

Which seems logical , also the sign- as further we go away from e^(pi/2) by infinite tetration, we get closer to -I, or +I, so increase in I+dI should lead to decrease of 1/2d(pi). In the end, we have just +-I, pi/2 has vanished. Comnig back from both -I, +I ovr infinite number of hyperdimensions, I vanishes, pi/2 is fully achieved.

Interestingly, e seems to fall out from dI based math in open form. But that has to be checked further. e appears only when integrals over certain amount oif dimensions are taken, meaning that e is variable as well, but depends on number dimensions included in integration over dI, so

e =e (function of number of hyperdimensions over which integral is taken).

Since e usually relates to time, one can see, that in individual hyperdimension there is no time. Time only appears when integrals over hyperdimensions are taken, and that is imaginary or perpendicular, if You like, time, if integration is finite.

Also, it seems obvious that lnI has 1 value on each dimension I-dI, I-2dI (may be ) etc , so there is no multivalued functions anymore- each of the values correspond to different hyperdimension.

The question of infinte values of hyperdimensions also arises when taking x root of y. There has to be infinite number of roots if x is real, not rational, even more so if it is irrational or transcendental, or imaginary.

if self root of i leads to e^pi/2, that means that either e^pi/2 has infinite hyperdimensional values, or, it is by itself a hyperdimensional number.
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#59
Dear Ivars!

Concerning the following:
Ivars Wrote:........ I am deliberately trying to find a true imaginary infinitesimal, not i *epsilon, sorry, not i *dx.
Well if we begin with i as infinitely rotated into higher dymensions length (e^pi/2) so that -i = h(e^pi/2) is a hypervolume of an infinite hyperdimension. OK? Totally unclear , no physical content, but that is what it could be.

Sorry, Ivars. I am trying to go "non-standard" since a lot of time but, this time, I again cannot follow you. We must agree on something, before proceeding further. Letter "i" means a number [the sqrt(-1), which actually is two numbers {-i,+i}, because (-i)^2 = (+i)^2 = -1]. Right? If you write Pi meaning the Greek letter Pi = 3.1415..., which is also a number, then the expression that you are mentioning could be: e^(Pi*i/2) = i and, therefore: e^(-Pi*i/2) = -i. So, you should clarify what do you mean by simple "p". Nevertheless, I cannot see e^(Pi*i/2) as a "dimension" and don't understand what stands for "e^pi/2". Moreover, what do you mean by "infinitely rotating around a dimension length" ?

Ivars Wrote:I know i is considered to be constant, but I do not see why that has to be accepted. i, pi, e are just symbols, and their content is in fact, infinite of unknown since they have no physical analoques.
di = 0 is an axiomatic assumption that works well in math. The same for d(pi)=0. Fine. And than we have Godels law and happily watch exponential growth and specialization of mathematics because we have limited ourselves to the usage of well defined symbols and application of well defined operations.The number of finite symbols and operations will grow infinitely.
... I is imaginary unit and equals infinite hyperspace volume. dI is a change of dimension of this hyperspace by one while still close to infinity. Very ill defined-no understandable content as infinity is involved.

You mean Imagination Power? Completely against Socialist Realismus ??!! ; - )

Ivars Wrote:If we than remove one lenghts e^pi/2 , back tetrate, so to say, what will happen with that hypervolume? Will it stay the same? or will it be one hyperdimension smaller?

Sorry, Ivars. I should like to suggest to stop these discussions here, please. Personally, I need a lot of time for preparing my old brain to receive these concepts. Let us try, for the moment, to concentrate us on tetration and on its conjectural uniqueness and smootheness. Otherwise, if we don't succeed to do so, unfortunately, we (at least ...I) have to give up.

Giamfranco
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#60
Dear Giamfranco,

GFR Wrote:Sorry, Ivars. I am trying to go "non-standard" since a lot of time but, this time, I again cannot follow you. We must agree on something, before proceeding further. Letter "i" means a number [the sqrt(-1), which actually is two numbers {-i,+i}, because (-i)^2 = (+i)^2 = -1]. Right? If you write Pi meaning the Greek letter Pi = 3.1415..., which is also a number, then the expression that you are mentioning could be: e^(Pi*i/2) = i and, therefore: e^(-Pi*i/2) = -i. So, you should clarify what do you mean by simple "p". Nevertheless, I cannot see e^(Pi*i/2) as a "dimension" and don't understand what stands for "e^pi/2". Moreover, what do you mean by "infinitely rotating around a dimension length" ?

The fact that -i , +i = integral over certain infinite number of hyperdimensions of dI in no way contradicts the fact that -I*+I = -1.

It just adds internal structure to I. No other changes in normal mathematics.

Quote:You mean Imagination Power? Completely against Socialist Realismus ??!! ; - )

Sure. Socialist Realismus has proven to be a dead idea both in theory and praxis. I have lived in THAT for 27 years, few of my relatives were killed/sent to Siberia in old days, and if You think it is better to live in s..t because also democracy causes pain to most people, than I prefer that pain of making choices. But that is outside topic, let us not go deeper.

Ivars Wrote:If we than remove one lenghts e^pi/2 , back tetrate, so to say, what will happen with that hypervolume? Will it stay the same? or will it be one hyperdimension smaller?


This was just a geometric interpretation of h(e^(pi/2) = -i given by Andrew, which I turned backwards. What I am asking is by what route do we reach hypervolume -i from moving length e^pi/2 into consequently higher dimensions and with each iteration how much closer to -i we get? And what if we start backwards, by (1/i)^i, from hypervolume -i ( or +i on another branch?) . How many selfroots of i should we have? How many x roots of y shall we have if x is real? irrational? transcendental?

"Stop the discussion"- I admit I was being little provoking here, but how should I stop it? I can not discuss alone.

For me it is the most important question mathematics should answer- how to go around, reverse Godels incompleteness theorem? And I see the answer in infinite tetration and adding illdefined hyperdimensional structure to things like I, pi may be more. And this has relation to physics very very much, to avoid building up more and more theories that makes understanding of whole impossible.

Ivars
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