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( 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.

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( 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.