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Exploring Pentation - Base e
hej GFR

My physical logic tells there has to be also superslow operations- perhaps inverses of fast operations- whose asymptotes in one direction might be complex, of a kind 1/2+ imaginary part.

Is it possible to define such inverse tetration, pentation, etc and calculate asymptotic values?

In that sense, zeration would not be the last in the row,or where would be these slow ones placed- as fractional oprations between 1 and 0, or negative operations?

I think that the infinite (contable) hyper-operation hierarchy y = b[s]x can (... probably!) be extended also to the ranks lower than zero. Nevertheless, please wait for the next thread on zeration. Concerning fractional (... rational) ranks, there is a possibility to define an operation between addition and multiplication (s=0.5 ?) apparently justified by the Gauss' Arithmetic-Geometric mean. We shall talk also of that. All this needs careful analysis and very precise demonstrations.

All non-commutable hyper-operations imply two inverses (the left- and the right- inverse operation), that we can define as belonging to the root and to the log types. This is the case of all integer ranks s >= 3 (exponentiatoion).

For the moment (...), we know that at ranks s=1 (addition) and s= 2 (multiplication), both commutable, these two inverse operations coincide.

jaydfox Wrote:Note that if we flip this graph about the line y=x, we'll see the pentalog. There will be a logarithmic singularity at about x=1.850354529. We can calculate the base of the logarithm by dividing the differences of two consecutive pairs of integer pentates. Going out to -100 iterations, This yields a value of about 6.460671295681839390208370083.

Just numerology: Can it be :

pi*(e^(1/e))^2 = 6,55670879
and inverse 0,152516........?

I have plotted all 8 "spirals " of the form 4 : ( +-t^+-1/t), 4: (+- 1/t)^+-(t) in polar coordinates and there are certainly interesting forms , crossing points, and regions.

I do not know how to get image in here, but it is in the atachment.

Attached Files Image(s)
I have plotted now Second order spirals in polar coordinates of type:

: ( +-t^+-1/t)^ (+- 1/t)^+-(t)

: (+- 1/t)^+-(t) ^( +-t^+-1/t)

It seems many of them repeat each other, so only 8 different ar left(may be I missed a few, quite labourious task)-see attachment.

In the limit as t-> infinity, they all cross at point with an angle phi= arctan(1) on a unit circle.

So if y would have been imaginary axis, the coordinates of crossing point would be

e^i = cos1+i*sin1.

Attached Files Image(s)
And a very strange spiral from plot in polar coordinates of:


It splits in 2 unconnected spirals at t=4,82*Pi in a strange way- at angle atan -1 there appears a connection from inner spiral to outer that continues the angle line, at some angle degrees to the unit circle. Then as t goes little above 4,82*pi, the program stops to draw this line and starts to drive a spiral downwards roughly paralel to unit circle, certain distance away. This can be seen in second attachment.

The argument in first attachment goes form 0 to 5pi.

One of them goes into unit circle quite exactly after crossing itself at arctan 1, reaching maximum and joining back unit circle approx at arctan-1?, while other just starts out of nowhere at x=-1,01, y=0,64... Can it be a bifurcation? or is it a software mistake?

The splitting spiral can be seen in attachment. If t is increased, in limit t->infinity the second spiral goes to some limit away from unit circle.
If plotted as y=f(x) in normal coordinates, my software shows discontinuity at x= 4,81976 *PI which is strangely close to

e^e = 15,15426224

But also to e^(pi/2) = 4,810477.

Arithmetic mean between them is 4,81744.. very close to approximate value i got. So is geometric mean.AGM seems to be 4,817112.....

It is not quite there, but I have only used 5 function of form (t^1/t), (1/t)^t.

Perhaps if more is used, or, if my sofware is mistaken, values will fit better?

The previous post has been totally reedited, with very intersting results maybe connecting

e^pi/2 =4,81..and e^e /pi = 4.82...via bifurcating spiral of a very long algebraic formula (5 selfroots) in polar coordinates.There are pictures as well, in attachments.

I tried few more spirals of the same type. When signs are varied at t in any place, they display interesting behaviour -some disapper, become point at coordinate beginning, few become scattered points in certain regions of argument t. For example this:


Is having Integral = 0 until 0,322Pi, then points are appearing which oscillating between close to 0 and close to 1 , and it stops at 4,4*pi. So software tries to sum these oscillating values , givin raise for the integral in the region up to 4,4*pi . Then it stops growing.

In polar coordinates(spiral) it has a point at origin and scatered points along segment of spiral.
May be my software can not calculate properly such long nested operations.
And this with 2 minus signs in the middle just makes unit circle in polar coordnates with fine structure ( if zooomed )starting from t=4,414689*pi which can be seen in integral jump.


Fine structure will be attched in next post.


Attached Files Image(s)

So here fine strucutre of Unit circle for:


I am not sure if this is even relevant, or artifact, but spirals are getting even stranger, the one with formula:


resembling snail shell. Extra t+5 in the end added straight lines away from center to bifurcation. All spirals have SPLIT patterns, are very sensitive to max value of parameter t, and go through point with coordinates ( sin1;cos1).
I mentioned Karman in one of file names as breakdown of smooth lines resembles Karman vortex sheets after an obstacle.

Attached Files Image(s)

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