Hmm; it seems impossible that the following was not yet discussed in our earliest tetration-discussion, but I can't remember and don't have an idea, where in our threads I possibly could find this. It just occured to me yesterday in a bar nearby where I tried to vizualize the spiral form of trajectory for integer-height-tetration in terms of the log-polar-representation on an envelope, with a beer at side...

1) having a real base, say b=exp(1), we assume, that beginning at a real value for x0 some positive integer or real iteration-height h gives some value x1, again on the real line.

So, if we fix some point x0 = 1.2 then for the continuous interval for h, say h=1..2 we get a continuous interval for x1, say x1=3.32 .. 27.66 . Or if we take x0=-4, then for h=1..2 we assume the continuous real interval for x1 of about x1=0.018 ... 1.0184

2) For complex values x0 the trajectory of the integer-height iterates x1,x2,... can have the shape of a spiral. Intuitively we would assume, that fractional height iterates for a continuous interval of heights h=1..2 fill that shape with a continuous curve, which roughly follows that spiraling shape.

3) There are complex values in the first quadrant, whose integer height iterations occurs in the second, and also in the third and forth quadrant. For instance

[h=-1 x_=1+0.5*î // update: I forgot in the first writing of this msg that the mysterious x0 below was in fact x1 iterated from that x_ , sorry]

h=0 x0 = 2.38551673096 + 1.30321372969*I

h=1 x1 = 2.87262925108 + 10.4780327918*I = exp(x0)

h=2 x2 =-8.74880093578 - 15.3675936421*I = exp(x1)

By (2) we expect, that the positive-real iteration curve from x0 to x2 is continuous and thus must cross the real axis. This happens for some fractional height h=1+µ ; so there exists some purely real iterate x_h when we begin at x0 and iterate continuously using base e. Also iterating further continuously with real heights we arrive at x2, which is a complex value (in the thirs quadrant).

But this contradicts (1), where we assume, that a positive real-height iterate of a real value x0 leads to a real value x1.

[arrgggh]

The evening is gone, the morning has come...still I've no other idea than that of a mindbreaking contradiction...

1) having a real base, say b=exp(1), we assume, that beginning at a real value for x0 some positive integer or real iteration-height h gives some value x1, again on the real line.

So, if we fix some point x0 = 1.2 then for the continuous interval for h, say h=1..2 we get a continuous interval for x1, say x1=3.32 .. 27.66 . Or if we take x0=-4, then for h=1..2 we assume the continuous real interval for x1 of about x1=0.018 ... 1.0184

2) For complex values x0 the trajectory of the integer-height iterates x1,x2,... can have the shape of a spiral. Intuitively we would assume, that fractional height iterates for a continuous interval of heights h=1..2 fill that shape with a continuous curve, which roughly follows that spiraling shape.

3) There are complex values in the first quadrant, whose integer height iterations occurs in the second, and also in the third and forth quadrant. For instance

[h=-1 x_=1+0.5*î // update: I forgot in the first writing of this msg that the mysterious x0 below was in fact x1 iterated from that x_ , sorry]

h=0 x0 = 2.38551673096 + 1.30321372969*I

h=1 x1 = 2.87262925108 + 10.4780327918*I = exp(x0)

h=2 x2 =-8.74880093578 - 15.3675936421*I = exp(x1)

By (2) we expect, that the positive-real iteration curve from x0 to x2 is continuous and thus must cross the real axis. This happens for some fractional height h=1+µ ; so there exists some purely real iterate x_h when we begin at x0 and iterate continuously using base e. Also iterating further continuously with real heights we arrive at x2, which is a complex value (in the thirs quadrant).

But this contradicts (1), where we assume, that a positive real-height iterate of a real value x0 leads to a real value x1.

[arrgggh]

The evening is gone, the morning has come...still I've no other idea than that of a mindbreaking contradiction...

Gottfried Helms, Kassel