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Tetration and selfroot as a topological transformation of CP1 into itself
#1
I have been thinking a lot lately...

I would like to share this:


Take a (topological) circle ("line" in CP1) with one point missing (point at "infinity") , decide that there is inside and outside of that circle and twist it by Pi so that to get 2 parts.

So You get 2 smaller circles, joined, likes this: O -> oo - and there we have 2 parts of something continuous out of 1 something with 2 ends meeting in the same point which WAS point at "infinity" before transformation we performed!

Then , the new thing oo is totally continuous-no gaps in line ( it looks like sign of infinity) . There is a double point where the line crosses itself . What was inside before of circle is now outside of 2 circles, what was outside 1 circle is now inside both circles because if we twist the circle the line forming it also twists in oposite direction, turns. So by this twist, the 1 "space" outside the line - let us say imaginary - is divided into 2 "spaces" inside.

Imagine this topological circle without 1 point is CP1- meaning it fills it all , there is no space, no distance, but it has (this projective "line" ) a topology of a circle without a point (=point at infinity) . Somehow that projective line manages in CP1 by twisting internally ( like an internal change happens in the line, of topological kind) become the analogy of O-> oo in euclidean space model I described- so somehow it gets turned inside out while there is no place to do it in CP1, from our point of view.

I think that this oo is truly continuos, while O is not.

Then, if we have a mathematical operation that does exactly this- takes circle with point at infinity in proj space CP1 and turns it in continuos line via twist, we have the ability to transfrom discrete ( with gaps ) into continuos ( no gaps) ,and back.

The biggest problem ( and only, in fact) is to transform the point at infinity in the circle (which is CP1) EXACTLY into the DOUBLE meeting point of twisted loop. So the "infinite" number characterizing the point at "infinity " on CP1 has to be transformed in - not a point, but a double point in the place where oo circles cross each other.

So we we need transformation that sends INFINITE number (not infinitely large, but infinite) characterizing point at infinity of CP1 to DOUBLE point in the same CP1, changing the topology of CP1, while mapping all points from 1 line to all points of 2 lines. Map 1 value -> 2 values.

And , we also need that ALL circles with points at infinty in CP1 (basically, the space CP1 itself) transform into these double loops (topological, inside CP1 "line") .

That is easy: Double point in CP1 is +-I, so called pair of circular points where all circles cross in CP1.

The transformation is tetration and selfroot:

Tetration of gap point (point at "infinity") e^pi/2 ( that is the number characterizing, assignable to the point at infinty of CP1) = double point +- I ( physically, that double point in CP1 is a spinor, which has consequences for physical application of infinite tetration)

Self root of double point +- I = I^(-I) = -I^(1/I) = e^(pi/2) = point at "infinity" in CP1.


So , tetration ( and self root) is a TOPOLOGICAL transformation of point at "infinity " in CP1 to pair of circular points in CP1, and it is the same as the twisting operation I described, with point at infinity ending exactly in the middle of 2 smaller "loops" (again, they are in projective space) , or, turning of projective line inside out via point at "infinity" to make it into CP1 with 2 circular points.

The operation of tetration /self root thus as whole may be an operation sending CP1 into ITSELF by radically changing topology inside this NO METRIC NO ANGLE NO DISTANCE NO SIZE space.

Are they called automorphisms or something like that? I am not sure.

Of course, based on this operation in CP1 1point-> 2 points , 2 points-> 1 point it is possible also to see what tetration does on Riemann sphere or with Riemann sphere. Also, there has to be analoguos topological transformations for HP1 (quaternion projective line) , OP1 (octonion projective line) sending 1 point into correspondingly 3(4) and 7(Cool values, or tripple ( quadrupple) and 7-fold(8-fold) points of topological jungle inside HP1, OP1.

Perhaps higher operations.

There has to be more extensions of this geometric analogy or essence of tetration, since it opens up a lot of areas.

Excuse me for weak mathematical formalism.
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#2
I have just realized what I have said above:

SO important to make connection with ususal math since it helps when many things are everything is calculated already:


In tetration, we are dealing with Topological Reordering of Real number line (real number line is not quite like projective, since it has 2 infinities- + and - BUT : the infinity of CP1, Complex projective line is ALSO a double point- so spinor - so +- oo) to istself . Meaning there can be either finite, or infinite number of ways to transform it , reorder. Imaginary numbers , quaternions etc are needed to describe these reorderings.

Now, since INFINITY (in CP1) is a spinor +- oo, what is than e^pi/2?

Complex projective spaces have a circle S1 1-dim-sphere at infinity
So e^pi/2 is a parameter of that circle S1.

What parameters do circle has?

Area and Radius and circumference.

If area is e^pi, what would be circumference dimensionally? =e^+-pi/2. Radius?

Little tricky, but it is e. (or, still its e^pi/2 since than tetration would transform spinor "radius" to spinor "radius").

So: e^pi is an area of circle at infinity of CP1, e^+pi/2 -circumference, e - radius (or e^pi/2-radius, circumference I can not say right now- it may have no circumference at all).

What is +-Pi/2 then? The imaginary Dimension of CP1 (inside the line) . The initial twist of line .It seems, this TWIST is concentrated at point at infinity. It could be that the point at infinity is the most general idea we have abou the INSIDE of a line CP1.

So inside Real Number line, dimensions of space become imaginary, and is thus related to the usual ones we use.

CP1can be seen as a combination of 2 RP1 spaces with infinities +oo ( +Pi/2 imaginary dimension) and -oo ( -Pi/2 imaginary dimension) .


Also, I wanted to mention, that continuous topological deformation of real number line via tetration leads to states in between full automorphisms (full twist e.g I described ) that are fractal iterations of topology of real number line or CP1. These fractal states of real number line achievable via CONTINOUS deformation of it ( INCLUDING infinity!!!) have

NON-INTEGER DIMENSIONS of SPACE.


Best regards,

Ivars
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#3
And here I add some supporting information for what I wrote previously from Tony Smith that he e-mailed to me about the above transformations and ideas: (his webspace is quoted a lot, by John Baez etc) http://www.tony5m17h.net/1TSphysics.html#clifvodou

Quote: Spheres and projective spaces can be thought of like this (I hope I have got this right,
but could be making some errors in notation or thought):

n-dim sphere Sn = n-dim real space Rn plus {point at infinity}

n-dim real projective space RPn = n-dim real space Rn plus { RP(n-1) at infinity } =
= Rn plus R(n-1) plus { RP(n-2) at infinity } =
= Rn plus R(n-1) plus R(n-2) plus R(n-3) plus ... plus { RP0 = two points = +1 , -1 at infinity }

n-dim complex projective space CPn = n-dim complex space Cn plus { CP(n-1) at infinity } =
= Cn plus C(n-1) plus { CP(n-2) at infinity } =
= Cn plus C(n-1) plus C(n-2) plus C(n-3) plus ... plus { CP0 = S1 = circle at infinity }

n-dim quaternion projective space QPn = n-dim quaternion space Qn plus { QP(n-1) at infinity } =
= Qn plus Q(n-1) plus { QP(n-2) at infinity } =
= Qn plus Q(n-1) plus Q(n-2) plus Q(n-3) plus ... plus { QP0 = S3 = 3-sphere at infinity }

My point is that the
real projective spaces have a two-point { +1 , -1 } structure at infinity
so that does look like spinor + - structure or a discrete Z2 2-element group at infinity and
complex projective spaces have a circle S1 1-dim-sphere at infinity
and since S1 = U(1) you can get a U(1) continuous symmetry group at infinity

(like electromagnetic U(1) gauge group)
and
quaternion projective spaces have a 3-sphere S3 at infinity
and since S3 = SU(2) you can get a SU(2)continuous symmetry group at infinity
(like weak force SU(2) gauge group)

To get the color force SU(3), you have to go to octonion projective spaces,
which (due to non-associativity) stop at OP2,
but you do have
OP2 = O2 plus { OP1 at infinity } =
= O2 plus O1 plus { OP0= S7 = 7-sphere at infinity }
so
octonions give you a 7-sphere S7 at infinity
Although S7 is not a Lie group, it naturally expands to Spin(Cool = S7 + G2 + S7
and G2 contains SU(3)
so that you can get a SU(3) continuous symmetry group at infinity
(like color force SU(3) gauge group)
and
furthermore if you look at Spin(Cool as a gauge group,
you can get (as I do in my E8 physics model) both Gravity and the Standard Model.

Tony

In terms of Lie Groups:

Quote:As to what I wrote in terms of Lie groups,
I hope this helps and does not have too many mistakes:

spheres Sn = SO(n+1) / SO(n)

real projective RPn = SO(n+1) / O(n) = SO(n+1) / SO(n) x {+1,-1}
note that O(n) is a double cover of SO(n)
which double cover comes from the {+1,-1} structure at infinity of real projective spaces and is geometrically related to the antipodal map between two hemispheres of a sphere.

complex projective CPn = SU(n+1) / U(n) = SU(n+1) / SU(n)xU(1)
and CP1 = SU(2) / U(1) = S3 / S1 = S2 by the Hopf fibration S1 -> S3 -> S2

quaternion projective QPn = Sp(n+1) / Sp(n)xSp(1)
and QP1 = Sp(2) / Sp(1)xSp(1) = Spin(5) / Spin(3)xSpin(3) = Spin(5) / Spin(4) = S4
where Spin(n) has the same Lie algebra as SO(n)

For octonions:
OP1 = Spin(9) / Spin(Cool = S8 which is also S15 / S7 by the Hopf fibration S7 -> S15 -> S8
and
OP2 = F4 / Spin(9)

If you enlarge F4 by
complexifying F4 you get E6
quaternifying F4 you get E7
octonifying F4 you get E8

Tony

And about automorphism groups of projective spaces:

Quote:Here are some remarks about automorphisms of structures at infinity:

spheres - only one point at infinity - the only automorphism is the identity 1

real projective spaces - {+1,-1} at infinity - automorphism is Z2, the 2-element group

complex projective spaces - 1-sphere S1 at infinity - automorphism group U(1) = S1

quaternion projective spaces - 3-sphere S3 at infinity - automorphism group SU(2) = S3

so for those, the structure at infinity can be given group structure so that
it looks like its own automorphism group

For octonions,
with an S7 at infinity, the automorphism group is not S7 itself because S7 is not a group,
but the S7 expands naturally to Spin(Cool = S7 + S7 + G2
where G2 is the octonion automorphism group
You might say that the expansion of S7 occurs because
the automorphism group of the octonions G2 is 14-dim and so bigger than 7-dim S7,
so that S7 has to expand by G2 to get group structure,
and then that process brings in a second S7 to produce S7 + G2 + S7 = 28-dim Spin(Cool.

Tony

not that I know it all, but suddenly it becomes very easy since we have tetration/selfroot ( and more) and their geometrical interpretation which EXACTLY fits the things the above groups, algebras, spinor spaces, covers describe.

From here, its a mere question of technique to develop both hyperops and understand and VISUALIZE complicated symmetries in space.

Tell me what You think about it, please.

Ivars
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#4
To put it very shortly:

Tetration is a transformation between Euclidean 1D Real Number line and 2-D spinor spaces.
Self root is a transformation between 2D spinor (null-plane, minimal line) space and Euclidean Real number line.

So this is transformation ACCROSS geometries.

Higher hyperoperations and their inverses perfmorm transformations between other geometries ( conformal, for example) and Euclidean.

Non-integer hyperoperations ( whose RANK is non-integer) correspond to transformations from e.g Real number line to a space which has fractional, in between geometry - say geometry in between Eculidean and 2-D spinor space for tetration/selfroot.

It may be called a transitional geometry, transitional state of Geometries, may be transitional non-integer dimension.


Ivars
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#5
And this twisting in and out of (spinor=null) space into Euclidean and back.. is spin. Twisting accross geometries is spin.

If we want to measure it in Euclidean space, observe, we BRING it to Euclidean space, so it is always aligned.

So the numerical value of h is hidden neither projective space, nor Euclidean, but in space of all geometries, and how this space represents numbers.

This twisting between geometries is also responsible for QUANTIZATION as such ( see also Regge Calculus with its strange fit to gravity).

Conformal geometry seems to be the geometry as if the twists are observed from outside. In Conformal geometry, 4 dimensions must play than a fundamental role (and they do, see tetracyclical and pentaspherical coordinates) , as twisting happens in 4 cycle between other geometries.
Null -Projective- Euclidean (affine)- Projective(?) - Null -etc...

So in Conformal geometrical space, tetration would be continuous. Perhaps.

So the generalization of logarithm I mentioned elsewhere should work as the correspondending values will be found in different geometries.
Perhaps ( I am not sure) generalized logarithm can be found as a continuous function in Conformal 5D (projective) space.

Also... Non-integer and imaginary, negative operations [x] would be linked to position in a twist between certain geometries. This operation count must have 4 cycle, as I indicated somewhere else in this forum, and may have link to alpha constant ( also I calculated it from Andrews pentation asymptotics) .

Non- integer values has to mean that operation happens BETWEEN geometries.



Ivars
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