02/20/2020, 01:24 PM

Consider the equation z^z = z.

This is fascinating because it makes the sequence ( power tower or tetration base z )

z^^n 1-periodic in n.

Maybe you think now that therefore it must be within the shell-tron.

Or maybe you think it is on its edge.

Let’s examine.

z = z^z.

1 , -1 are solutions.

And maybe some accept 0. But imho 0^0 = 1.

These solutions ( bases) are problematic for tetration to say the least.

Please correct me if I’m wrong about that !!

But !

There are nonreal solutions as well !

And they are outside the shell-tron !

Even more surprising the derivative at their fixpoints is not a half-rotation !!

So counter-intuitive I might say.

Im not sure how many have repelling or attracting fixpoints.

I think none are parabolic.

That is interesting and surprising.

The idea of cyclic orbits emerges.

This reminds me of the base 1.7129 i.

But this is different.

The smallest solution is ( or its conjugate )

About

2.86295 + 3.22327 i

I am fascinated by it.

Regards

Tommy1729

This is fascinating because it makes the sequence ( power tower or tetration base z )

z^^n 1-periodic in n.

Maybe you think now that therefore it must be within the shell-tron.

Or maybe you think it is on its edge.

Let’s examine.

z = z^z.

1 , -1 are solutions.

And maybe some accept 0. But imho 0^0 = 1.

These solutions ( bases) are problematic for tetration to say the least.

Please correct me if I’m wrong about that !!

But !

There are nonreal solutions as well !

And they are outside the shell-tron !

Even more surprising the derivative at their fixpoints is not a half-rotation !!

So counter-intuitive I might say.

Im not sure how many have repelling or attracting fixpoints.

I think none are parabolic.

That is interesting and surprising.

The idea of cyclic orbits emerges.

This reminds me of the base 1.7129 i.

But this is different.

The smallest solution is ( or its conjugate )

About

2.86295 + 3.22327 i

I am fascinated by it.

Regards

Tommy1729