# Confusing pictures about tetration !? [closed]

On the webpage http://tetration.org/Tetration/index.html,

We are supposed to get an explanation of tetration, whatever that means exactly.

In particular I feel the pictures are not well explained.

The first two pictures show “ tetration “ and the last a “ Julia set of 2^z “.

However what is meant by tetration ?? Tetration has many parameters , interpretations solutions etc.

My guess is the pictures are related to z^z^z^... where the base is z and the starting value is z. And then we consider z^^n. And we get limits , double limits , triple limits etc or no ( finite ) limit at all.

And then we color them accordingly.

However , even if that is the case , it should have been stated clearly. It might be something else ?

Also , there are 2 pictures , slightly different. And labeled mysteriously : “ by escape and by period “

What does that mean ??? Did I describe one of them ??

Also there are only finite colors.

And nothing about the pictures is explained , proven or even conjectured.

And the last picture is suppose to be a Julia set of an exponential function.

However , exponential functions are strongly chaotic and arbitrarily close to almost any point is a periodic point.

Hence for most bases I find it hard to interpret a Julia set.

Julia sets are well defined for polynomials and a handful transcendentals.

But without attracting points the Julia set of a transcendental entire function is “ fuzzy “.

Exp(z) has no attracting fixpoints for example and near any point is a periodic point, and the iterations are chaotic.

The pictures appear on other places of the website and are also copied to places like Wikipedia , the tetration forum or archiv. And many more. BUT also without explaining things.

Also I assume the domain is colored and not the range of z^z^...

Also I find the colors not even convincing ? All that connected green space ? All large imaginary are green ? Really ??

Or is that just an illusion and we get more structure when zooming out ?

My friend noticed that the equation

z^z = z does not lie within the shel-tron region ( I presume the red part in pic 2 ) ( the smallest nonreal solution z is not in the red. ) It might be true that every solution to

$$z^{^n} = z$$

( this is not a power on the LHS but a power tower of size n ) (For all n and z)

Might have its own “ color island “. I mean a one-to-one correspondence here.

Do not get me wrong, I appreciate someone maintaining a site about tetration. But after all these years it should have been made clear imho.

Hopefully you can clarify things.

• As the author of tetration.org I can clear up your questions. Just so you know, I'm moderately autistic, so if no one raises an issue, I may not understand a lack of understanding by others. – Daniel Geisler Mar 6 at 5:30
• Mick, feel free to contact me at daniellgeisler@gmail.com. Also I believe that like myself, you are a member of the Tetration Forum. – Daniel Geisler Mar 6 at 6:35
• Im not a member of the tetration forum. I just read there. – mick Mar 7 at 19:23

The first two pictures show “ tetration “ and the last a “ Julia set of $$2^z$$ “. However what is meant by tetration ?? Tetration has many parameters , interpretations solutions etc. My guess is the pictures are related to $$z^{z^{z^\ldots}}$$ where the base is z and the starting value is $$z$$. And then we consider $$^nz$$. And we get limits , double limits , triple limits etc or no ( finite ) limit at all. And then we color them accordingly.

(* Fractint code*)

TetrationM (XAXIS) {; z = pixel: z = pixel ^ z |z| <= 100000 }

Correct. The Tetration fractal goes back to Lee Skinner's Tetrate algorithm in the Fractint fractal generator. This was much better known twenty years ago. The 'escape' fractal is the exponential map analog to the Mandelbrot set. The 'period' exponential Mandelbrot has a complicated period structure that Fractint also generated.

... Did I describe one of them ??

Perfectly, you described the 'escape fractal'.

Also there are only finite colors. Yes as the Mandelbrot and other fractals use finite colors. z^z = z does not lie within the shel-tron region ( I presume the red part in pic 2 ) ( the smallest nonreal solution z is not in the red. )

Correct again.

The Period fractal is just the inverse of the escape fractal. A point will be black in one fractal and a color in the other. Fractint had a mode to quickly compute the period of forward orbits that didn't escape.

How about the chaotic nature of iterations $$2^z$$ ; how can we make a nice Julia set then ?

Your concerns about $$2^z$$ can be addressed by the Julia set of the exponential function $$e^z$$. The cover of An Introduction to Chaotic Dynamical Systems, by Robert Devaney has a Julia set of the exponential function. See Section 3.8 which is devoted to the topic. All the green area is period three.

• Thank you for your answer Daniel. I never heard of Lee skinner. I’m too young I guess. Although I know Newton haha. Maybe a link for mr Lee or would a google search be sufficient? Anyways apparently I described the escape fractal. But that also means I did not describe the other 2 pictures. So , how about the period fractal ? And of less importance but still ; how about the chaotic nature of iterations 2^z ; how can we make a nice Julia set then ? Basicly you answered about 35 % to be blunt. But if you could explain The periodic fractal. And maybe the Julia of 2^z ... then I could accept this. – mick Mar 7 at 19:10
• And maybe this thread implies that you should clarify the pictures of your website. Do not take my confusion or critisism too personal. I just think for the benefit of tetration that things should be presented as clear and simple as possible. Ofcourse be careful with editing your website. The websites contains great stuff too. The combinatorics as example. – mick Mar 7 at 19:16
• What do you think about the color island conjecture? I assume this is an unpublished question. – mick Mar 7 at 19:19
• See Tetration Forum for recent discussion relevant to your color island conjecture. – Daniel Geisler Mar 8 at 14:45
• Thank you for updating the answer. So apparently chaotic dynamical systems have fractals too. I was unaware of that sorry. Nobody ever told me. You gave a reference to a book. Does this imply it is too complicated for a short answer ? But my main issue remains that “ The Period fractal is just the inverse of the escape fractal “ does not clarify anything to me !? What is the definition of the period fractal ? I assume it is not defined as the inverse of another fractal ?? Also WHY is it the inverse ?? And lastly details ; what is the number of max iterations , what is the escape value ? 10^10? – mick Mar 8 at 21:48

We are suppose to get an explaination of tetration

is where you start to go wrong, I believe.

Look at the second paragraph of text on that page, before you even get to the pictures: the author clearly indicates that the purpose of the page is to explore "how can tetration be extended to complex numbers."

This page, then, assumes a comfortable grasp of tetration over the reals as background knowledge.