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I have tried to help the building of a tetration community for several decades, but I'm at a loss regarding the Tetration Forum. The Forum was put up several years after I put up Tetration.org, but it appears to be focused on providing alternatives to the material on my website. An early rallying cry on the Forum was that fixed points don't matter. But over time I see that is no longer feel it is true. I'd love to hear an explanation of why my tetration website is rated so high by Google, why Wikipedia uses chunks of my website and yet there is zero references to my site or any support for my work on the Forum.
People show up for a few years on the forum, take their best shot, and then disappear forever. I suspect the Forum in on its fourth generation of members. Maybe we can't agree on mathematical principles, but find human principles to agree on. Personally, I care much more about people than mathematics. I'm trying to deal with the amazing weirdness of Cisco Tetration Analytics. Before that is was Joshua Cohen's NYT Best Seller "Book of Numbers" where the largest company in the world is "Tetration.com".
I want to hear from individuals about how they and others in the Tetration community are doing. If no one cares, I can handle that because I'm retired. We can just step aside and let future generations worship the wonders of Cisco's Tetration Analytics platform.
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I'm really not sure why mainstream mathematics hasn't taken tetration more seriously. It's a fascinating subject, and feel like the necessary tools to do a deep analysis have been available for decades. Kneser's work over 70 years ago was done in the absence of computer assistance. Imagine how much more the maths community could have accomplished in the 1970's and 1980's! I seem to recall a few efforts in the 1990's (Peter Walker?), but nothing groundbreaking comes to mind.
I get that it might not seem as interesting as the Riemann Hypothesis or Poincare's Conjecture, but I see a lot of really esoteric maths stuff get published, which seems far less fundamental than iteration theory and tetration. I feel like amateurs like myself and others here have really had to step in and do the work, in the absence of highlevel math superstars.
I also feel like the amateur community has been reinventing the wheel over and over and over. On a personal level, it is somewhat satisfying for me to work on the problem off and on for years, and then to "rediscover" something that has already been discovered independently by several others before me. But the fact that several of us independent amateurs have to rediscover the same solutions over and over, speaks to how poor the published literature is on this subject. Even the scant available literature seems hard to digest.
I have been swamped at work for the last year on a large project, which has really stifled my ability to work on tetration projects on the side. Before that, I took several years off. I feel I am very close to some very cool results, which of course have already been discovered by others. I would love to collaborate with some of the others here (e.g., Sheldon or Henryk) to take my code and ideas and formalize it into a paper. Maybe even some sort of monograph that takes someone through a journey of analyzing various methods, with examples and less rigorous methods at the start, something that can be followed more intuitively, before getting into the more formal maths. I would love to do some small part to make this branch of mathematics more appreciated and understandable.
I can't properly explain how exhilarating it is to work on the problem for months, and then to have one of those Eureka! moments when some key part of the solution suddenly makes sense.
Anyway, I guess I'm rambling. I'm sorry you've become jaded by the lack of support in the community. For what it's worth, I hope you consider making your website available again as some point.
~ Jay Daniel Fox
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07/07/2019, 12:32 AM
(This post was last modified: 07/07/2019, 01:08 AM by Daniel.)
Thank you for your reply. If I come across intense it's because I try to be more than a researcher, I try and build community. I try and stay connected with Yiannis Galidakis and represent his insights and interests. Gottfield can vouch that every now and then I try and find out how people are doing. I'm looking at how to get some cool stuff happening like sponsorship for a tetration conference.
I'm really not sure why mainstream mathematics hasn't taken tetration more seriously. It's a fascinating subject, and feel like the necessary tools to do a deep analysis have been available for decades. Kneser's work over 70 years ago was done in the absence of computer assistance.
Note that Wikipedia's writeup for the fractional iteration of functions on the Iterated Functions page was material copied from my website without attribution. I argued with the editors, looked at starting an edit war, but attributed my website and left. That was five years ago.
My answer to f(f(x))=g(x) is more general and succinct by using the inverse of Bell polynomials
Code: f[0]=0;
max = 3;
Solve[Table[D[g[z] == f[f[z]], {z, i}] /. z > 0 , {i, max}],
Table[D[f[z], {z, i}] /. z > 0, {i, max}]]
Imagine how much more the maths community could have accomplished in the 1970's and 1980's! [b]I seem to recall a few efforts in the 1990's (Peter Walker?), but nothing groundbreaking comes to mind.[/b]
 1971 Began research on tetration
 1979 Studying complex exponential dynamics with TI49
 1983 Discovered superattractor, attractors, repellors, rationally neutral and irrationally neutral fixed points of the exponential map
 1986 Series of important discussions with Stephen Wolfram regarding tetration and the unification of discrete and continuous iteration
 1987 Hand calculation of the first five derivatives of hyperbolic tetration. Began studying complex dynamics.
 1991 Submission of my first paper to Wolfram as per our arrangement, but Wolfram had left academia and was busy with Mathematica
 1992 Research invalidated by the appearance of infinities when the multiplier is a root of unity.
 1994 Realized what I had thought was a flaw was an important successful prediction that the topology based Classification of Fixed Points was also able to be derived algebraically.
 1994 Published continuously iterated sine function on sci.math
I'm sorry you've become jaded by the lack of support in the community.
Given my tetration.org website is much higher rated in Google than the Tetration Forum and preceded it by several years, yet there is never any reference to my work on this site it feels like the Tetration Forum could change its name to "Anyone, but Geisler." I'm not trying to be an asshole, it's just disappointing to see many dozens of people who were capable of "discovering tetration", but were not smart enough to find their way to tetration.org.
FYI  I am very autistic and have an extreme case of visual synthesia. These two are in common with autistic savants. Participation in brainwave studies in the midseventies revealed that unlike the population at large, if I'm in the alpha state and someone gives me a simple math problem, I will remain in the alpha state. One researcher's response was to yell "that's impossible." I flunked second grade then discovered discrete calculus in third grade. I also could solve the simultaneous equations from chemistry in third grade. I'm trying to connect up with more advanced technology like MRI to map out my brain, but I think there is strong evidence that I am an autistic mathematical savant.
In 2002 I was visiting New Jersey, so I made a pilgrimage to Princeton University and their mathematics department at Fine Hall. The school was closed that day, but I enjoyed reading stuff on the office doors of my heroes from mathematics. I kept noticing another guy in the background. At the end, I sat for some lunch outside Fine Hall and the other guy was nearby. Finally, he came over to me and said he had noticed me around and wondered what my interest was. The guy was very odd because he kept punctuating our conversations with the claim that he didn't know anything about mathematics, and then would display a seeming knowledge of the entirety of mathematics. So I explained my personal interest in visiting Fine Hall with why I considered Princeton University to be so important. Then he dropped a bomb. He was the recruiter for Princeton's mathematics department. He claimed that my ability to work alone without formal education while rediscovering decades of mathematics meant that I was a mathematician like Ramanujan. Then he told me that the Princeton math department would like me to join their department as a student. Unfortunately being a transfer student nixed that idea.
Karma is people who rather have their work supported than a predecessor finds out no one wants to support their work either.
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07/09/2019, 08:44 PM
(This post was last modified: 07/09/2019, 08:44 PM by bo198214.)
Dear Daniel,
I can understand that it does not feel good to not getting the recognition you wanted. However to wallow in disappointment does not help you nor does it help the forum.
To be very direct: This may be the perfect time for you to face: that you are not the great mathematician with highly advancing contributions to the field, that you are not on one level with Ramanujan, to consider that you may just be an ambitioned, enthusiastic layman, fascinated by the topic ... like most of us on the forum. (Perhaps with more interesting brain patterns and Princeton encounters.) The performer is not in the position to enforce its applause (except for a tyrant perhaps ).
But back to the original question of how we are doing: I personally have withdrawn from the topic for some years already due to a total lack of time that I could dedicate. I am aware that this may also one factor that the intense discussions on the forum faded over time and I dont feel good about it. I totally share what jaydfox said about missing adoption in professional mathematics and rediscovering results due to a lack of introduction/overview what was found so far and what not yet was found. (Actually that was one of my new year resolution to prepare an easy overview, also to tease further research ... however I did not even begin  sigh).
And there was found a lot already by the community (so many different ways of calculation tetration (e.g. Andrew, Ansus, Gottfried, Sheldon, JayD), though mostly with no proof of convergence (even the recent papers by Kouznetsov and Paulsen, have no proof of convergence nor what it would converge to, they are purely numerical observations), uniqueness criteria, connection with LeauFatouflower, etc) but also so many open questions.
So I only can give my best, find some time, to work on an overview/current state and perhaps post it on the tetration wiki (after I update the software ...).
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07/09/2019, 09:44 PM
(This post was last modified: 07/09/2019, 10:05 PM by Daniel.)
Please focus folks. I don't have a problem with not receiving recognition, because I have received recognition. Well over five years ago material from my website was posted on Wikipedia's iterated function page under the fractional iteration header. The editors of the page will not allow the material to be removed even though they accept that the material comes from my website and that it is not peerreviewed. Feel free to straighten them out.
Let me be direct Trappman, from our previous interactions I believe that you despise me as a human being. What the Hell did I do to you to earn this? Let's talk tangibles. I'm not claiming that your website doesn't support my website, I claim that a priori no information from my website is discussed here. Go ahead and show me where my work was even considered on this site. Since my site ranks far higher than this site on Google and has been up for a dozen years it is only natural to question whether hidden politics is behind your website.
Your problem is to be the most important tetration researcher, my work must be discounted. You don't have the intellect to do it, so you stoop to character assassination. In the end, the truth will come out. So I've moved on to supporting researchers.
Trappman, I find your disregard for the Principles of Critical Thinking disappointing. You think you have the insight to peer into my mind and reveal what no one else including myself can see? This is a sad shallow move.
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07/10/2019, 12:51 AM
(This post was last modified: 07/10/2019, 06:57 AM by sheldonison.)
(07/09/2019, 08:44 PM)bo198214 Wrote: Dear Daniel,
I can understand that it does not feel good to not getting the recognition you wanted. However to wallow in disappointment does not help you nor does it help the forum.
To be very direct: This may be the perfect time for you to face: that you are not the great mathematician with highly advancing contributions to the field, that you are not on one level with Ramanujan, to consider that you may just be an ambitioned, enthusiastic layman, fascinated by the topic ... like most of us on the forum. (Perhaps with more interesting brain patterns and Princeton encounters.) The performer is not in the position to enforce its applause (except for a tyrant perhaps ).
But back to the original question of how we are doing: I personally have withdrawn from the topic for some years already due to a total lack of time that I could dedicate. I am aware that this may also one factor that the intense discussions on the forum faded over time and I dont feel good about it. I totally share what jaydfox said about missing adoption in professional mathematics and rediscovering results due to a lack of introduction/overview what was found so far and what not yet was found. (Actually that was one of my new year resolution to prepare an easy overview, also to tease further research ... however I did not even begin  sigh).
And there was found a lot already by the community (so many different ways of calculation tetration (e.g. Andrew, Ansus, Gottfried, Sheldon, JayD), though mostly with no proof of convergence (even the recent papers by Kouznetsov and Paulsen, have no proof of convergence nor what it would converge to, they are purely numerical observations), uniqueness criteria, connection with LeauFatouflower, etc) but also so many open questions.
So I only can give my best, find some time, to work on an overview/current state and perhaps post it on the tetration wiki (after I update the software ...). I think we all owe Henryk a thank you for putting in the energy to create and maintain this website. Someday, maybe when I retire, I can gather up some of the cool stuff here, some of which I contributed to, and maybe talk about what it would take to make it more rigorous (and maybe actually prove a few things????)
Also thank you to all the other folks on this forum with whom I have interacted.
I have occasionally crossed paths with you Daniel, I think more on mathstack, and I've certainly enjoyed your website, which is now relegated to the internet wayback machine .... However, as much as I liked your website, I wouldn't call it an exhaustive overview of the topic.
I am one of the folks Henryk describes, "ambitioned, enthusiastic layman, fascinated by the topic ..." who is mostly self taught, but weak on proofs, and even weaker on actually publishing anything. But hey, I like the fact that I've learned complex dynamics, and Peter Walkers nowhere analytic infinitely differentiable solution, and Kouznetsov's algorithms, and Jay and Andrew's matrix based slog's and studied their weird convergence, and Henryk's uniqueness sickle, and Paulsen's papers and Gottried Helm's Matrices, edit Kneser's solution, LambertW, Riemann mappings, 1cyclic mappings, parigp, end edit and lots and lots of other stuff, none of which I knew 10 years ago.
I guess 10 years isn't much next to the nearly 50 years that Daniel has been studying superexponentials...
 Sheldon
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Thanks for the feedback folks. There is more than one way to make an impact in mathematics. You can discover material in your research, you can be a scholar, you can write material for others to learn and you can support the researchers. As a human being, it doesn't matter what I discovered if I don't look after the wellbeing of others. So I'd like to understand how I can be of service to everyone here, including Trappman.
Do I have concerns? Yes, but not personal. I try being a friend to Yiannis Galanakis. He is someone that the whole tetration community owes. He is now working to complete his second attempt at a PhD in mathematics, so maybe an investment of sixteen years. I think it would be great for the brother to hear how much his help has done for us.
I do have issues about Cisco Tetration creating confusion. But since they wanted the name I think they should pay some dues, just as the people on this forum have. My weird idea is to get financial support for a tetration conference so that we can meet each other. Hopefully, the researchers on the Tetration Forum can be publically accepted as nerds.
I can give folks feedback to questions like why professional mathematicians don't support research into tetration. I have the perspective of working with Stephen Wolfram since 1986.
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07/10/2019, 02:25 AM
(This post was last modified: 07/10/2019, 03:24 AM by Daniel.)
FYI  let's chill on the Daniel is butthurt because he isn't famous. I am moderately autistic and on retirement. I DON'T FUCKING CARE ABOUT BEING FAMOUS!!! I am much better off as an individual living a quiet life.
All together now; "autistic people have no need to give a damn." I do not need you, I do not need anyone and it is neurologically impossible for me to give a shit about being famous. I do care about being a good human being. My question is why is being a good person so difficult to recognize?
I serve as a spokesperson for the autistic community. Fifty years of tetration research got me a nice homeless camp by a nearby river for over a year. So there is some attitude. I'm sixtytwo years old. The life expectancy of autistic people is sixty, the unemployment rate is 90% and the main cause of death is suicide.
So my Princeton story upset some folks. Since no one understood where I was going, I'll spell it out. I have dozens of stories like Princeton. People can't understand why their research hasn't established tetration. My point is even if you were a mathematical autistic savant with half a century research experience, establishing tetration could still be far beyond your capability.
Back in 2002, I was recruited by the physics professor who was my mentor in the early seventies. I excitedly explained my work on extending tetration to the complex numbers, of unifying maps and flows. His response was, "I thought Poincaré proved that was impossible!" Our attempts to unify maps and flows are even more difficult than solving a Clay Prize. This is the very problem in dynamics that split mathematics into algebra and topology. To say I have extended tetration to the real and complex numbers is to say you succeeded where Poincaré couldn't. So many mathematicians consider tetration just talking crazy.
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At least I want to state:
The forum is a free place for exchange about Tetration related topics.
There is no hidden agenda and particularly no intend to avoid Daniels research.
And anyways, if you wanted your topics discussed here, then you just make a post. Its also a riddle to me how I am supposed to influence people not to talk about your stuff ... And  on another note  drifting into conspiracy theories makes it finally impossible to seriously discuss things.
