And a very strange spiral from plot in polar coordinates of:

(t^(1/t))^((((1/t)^t)^(((1/t)^t))^(((t^(1/t))^((1/t)^-t)))));

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

=4,827351*pi

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?

Ivars

(t^(1/t))^((((1/t)^t)^(((1/t)^t))^(((t^(1/t))^((1/t)^-t)))));

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

=4,827351*pi

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?

Ivars