share with others#plotted #functional_iterations

[Leo.W]

hey bodies I felt like to share some of plots of iterations with u *just for visual treatment(?no)
plz feel free to share if you have any interesting graphs (gifs are better!)


for \(exp_e(z)\): (due to some branch cuts there are 2 points in the right and some cuts in the left should be excluded)
p1: \(exp_e^t(z), t\in[-1,1]\)
p2: \(exp_e^{e^{it}}(z), t\in[0,2\pi)\)
p3: \(exp_e^{it}(z), t\in[-1,1]\)
also I have to say that does the iteration to an imag height always tortune and rotate the original function in some manner? at least the photos represent to me with such impression
All these iterations generated by Kneser's tetration.

for \(sin(z)\): 
p4: for real \(z\in[-\pi,\pi]\), plot \(sin^t(z),t\in[0,1]\)
Generated by fixed point method.

[bo198214]

The complex plots look a bit fuzzy.
But in any case very much appreciated!
However you should add which iteration it is!
In the case \(\exp_e\) I would guess it is the Kneser iteration.
For \(\sin\) I would guess it is regular iteration at 0, right?

[Leo.W]

The complex plots look a bit fuzzy.
But in any case very much appreciated!
However you should add which iteration it is!
In the case \(\exp_e\) I would guess it is the Kneser iteration.
For \(\sin\) I would guess it is regular iteration at 0, right?

Yeah rigority always wins  sadly if I make it more elaborate the gif will take e-memories like an apk for an app
all by wolfram mma, I forced a function to plot complex graphs but it'll still leave a mark at top right angle
I also had graphs for sin at 0(complex) tan at 0(complex) and e^x-1 at 0(complex) but in another pc, will append em soon

[bo198214]

I also had graphs for sin at 0(complex) tan at 0(complex) and e^x-1 at 0(complex) but in another pc, will append em soon

I thought, I am the one who makes a lot of animated pictures!
Looking forward.

[tommy1729]

much appreciated !

I wonder how different methods give different plots.
Maybe the differences are too small to see.

regards

tommy1729

[JmsNxn]

I also had graphs for sin at 0(complex) tan at 0(complex) and e^x-1 at 0(complex) but in another pc, will append em soon

I thought, I am the one who makes a lot of animated pictures!
Looking forward.

God I tried making animations and just got so god damned frustrated. I wish I knew how to use Sage better; but I couldn't get passed the syntax used to convert Pari-gp code into the Sage shell; and then run Sage's graphing protocol. I tried reading some breakdowns, but to no avail. Nothing helped. Especially because all the tutorials seem to be for the free to use Sage online app that seems to be the status quo now. I would love to make animated plots though. Ember made some animated plots of the beta method as you move \(b\) around.


Here are some graphs that are my favourites because they have cool fractals.

This is tetration base \(\sqrt[3]{3}\) with \(2 \pi i\) period; and centered at the singularities which happen at \(j+\pi i\). This is one of many "candy stripe graphs" I have

   

This is a similar graph, but we've changed to \(\sqrt[3+0.5i]{3}\)--still \(2 \pi i\) period.

   

This is a similar graph, but we're about \(b = e^{-0.3-i}\)

   

This one is really cool because it's for a period that looks like \(2\pi i/(1+i)\) and for base \(b = e^{1+i}\), so it gets pretty whacky. This one isn't a tetration, it's just the asymptotic solution.

   

And then the all-time disappointment. This is when we let the period be \(2\pi i/0.25\) and the base \(b = e\). You can see what sheldon called the clustering of zeroes and branch cuts near the real line. They look like little tornadoes or something. Ultimately led to the proof that the beta method isn't analytic anywhere in \(\mathbb{C}\) when the base is \(b=e\). The only hope would be to let \(0.25\) get smaller and smaller (where the coefficients do get more regular).

   


Also, I have hundreds of these graphs if anyones interested. I have about 1gb of tetration graphs

[Leo.W]

Also, I have hundreds of these graphs if anyones interested. I have about 1gb of tetration graphs

Totally interested! Love ya super cool graphs
your beta method looks very nice. Albeit its non-analytic-ness, it can further shows tons of info such as the almost-discontinuity of original kneser's tet_e and a structure, and beneath that it's super-cool

[bo198214]

OMG that's a real artwork!
Maybe you can make an NFT out of it and earn a lot of money!!!

God I tried making animations and just got so god damned frustrated. I wish I knew how to use Sage better; but I couldn't get passed the syntax used to convert Pari-gp code into the Sage shell; and then run Sage's graphing protocol. I tried reading some breakdowns, but to no avail. Nothing helped. Especially because all the tutorials seem to be for the free to use Sage online app that seems to be the status quo now. I would love to make animated plots though. Ember made some animated plots of the beta method as you move \(b\) around.

Not exactly sure what you are doing. But an animation basically just a sequence of plots.
Say you have a plot
a=complex_plot(....(t)......)
and then you want to make an animated picture out of it, then you just do

Code:

ani=[]
for t in sxrange(0,T,delta):
  ani.append(complex_plot(.....(t).....)
animate(ani)

sxrange goes in steps of delta from 0 to just below T.
This can though take a long time.

We also can make in interactive session - say next weekend. I have some experience with Sage.

[JmsNxn]

OMG that's a real artwork!
Maybe you can make an NFT out of it and earn a lot of money!!!

God I tried making animations and just got so god damned frustrated. I wish I knew how to use Sage better; but I couldn't get passed the syntax used to convert Pari-gp code into the Sage shell; and then run Sage's graphing protocol. I tried reading some breakdowns, but to no avail. Nothing helped. Especially because all the tutorials seem to be for the free to use Sage online app that seems to be the status quo now. I would love to make animated plots though. Ember made some animated plots of the beta method as you move \(b\) around.

Not exactly sure what you are doing. But an animation basically just a sequence of plots.
Say you have a plot
a=complex_plot(....(t)......)
and then you want to make an animated picture out of it, then you just do

Code:

ani=[]
for t in sxrange(0,T,delta):
  ani.append(complex_plot(.....(t).....)
animate(ani)

sxrange goes in steps of delta from 0 to just below T.
This can though take a long time.

We also can make in interactive session - say next weekend. I have some experience with Sage.

Lmao at nft!

It's actually not sage that's the problem. It's that I prefer to write my code in pari. And I'm having a lot of trouble at getting sage to read my pari gp code. Sage claims to have an onboard protocol to read pari gp code natively. But everytime I try to get it to work its just so god damned frustrating.

I can understand how Sage works, even though it's not my favourite kind of language (I tend to like C/C++ environments, not python java kind of environments). The trouble is, I don't understand how to make sage use a function defined within my pari-gp code; and run that function within the sage shell. I gave up trying to get it to work. I know it's probably something dumb I'm not typing correct; or forgetting to read some library or something like that. I'd love to get some help with it to be honest. I've always wanted to animate:

\[
\beta_{\lambda,\mu}(s)\\
\]

While we move \(\lambda\) or \(\mu\). These converge pretty damn fast, and the code is pretty damn fast (thank god for Sheldon, he wrote a couple speedups to my old code that made my code look like a mazda racing a ferrari, lol). Also graphing these is actually surprisingly fast. What's slow is graphing the tetration function, which I'd still like to animate, but that'd probably take a week of processing time, lol.

Also, I have hundreds of these graphs if anyones interested. I have about 1gb of tetration graphs

Totally interested! Love ya super cool graphs
your beta method looks very nice. Albeit its non-analytic-ness, it can further shows tons of info such as the almost-discontinuity of original kneser's tet_e and a structure, and beneath that it's super-cool

I'll post more graphs. I'll compile some of the greatest hits, lol.

[JmsNxn]

So this is a fun one, that I think is extraordinarily interesting. This is related directly into "semi-operators" or "half-operators" as we can call them. If we take:

\[
x \langle s\rangle y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)
\]

Then this function interpolates \(+,\times,\exp\). And nearly satisfies the equation:

\[
x\langle s \rangle (x\langle s+1\rangle y ) \approx x \langle s+1 \rangle (y+1)\\
\]

I always wanted to make an animation of these continuous interpolations, because they have all the properties of half exponentials.

Anyway, here is \(3 \langle 1.9\rangle y\), which is almost periodic like \(3^y\), but just misses the criteria. These iterations are done entirely using the repelling fixed point iteration (about \(4\) rather than \(2\)). 

   

And to imagine what the animation would look like, here is \(3 \langle 1.5 \rangle y\):

   

This one is an example of a super function of exponentiation when the base is \(1/2\) and, yes the period is \(2 \pi i\). You can see a lot of chaos, but in this instance, this function is pretty much holomorphic everywhere--save a few singularities and branch cuts, lol. And yes, IT IS TETRATIONAL. The thing is, there are countably infinite \(x_0\) to choose as a normalization point .

   

This one is a graph that is trying to construct Kneser's iteration. This is done using the conjectured formula:

\[
\text{tet}_{K}(s) = \lim_{n\to\infty} \log^{\circ n} \beta_{1/\sqrt{1+s+n + s_0}}(s+n+s_0)\\
\]

This graph is still incorrect though, and isn't converged perfectly yet; as this was a low recursion graph (still probably took me 3 days to compile, lol).

   

This is another graph which attempted to show the regularity of the \(\beta\) method when you let \(\lambda \to 0\):

   

Despite the niceness of this graph, we have fairly poor Taylor series data. Not disproving the result, just slow. But it looked a good amount like Kneser's Taylor data. Just converging poorly.