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For my retirement I'm only going to do what is fun mathematically. I had wanted a way to share the beauty I saw in the mathematics of tetration. Then I came across Fractint and created the fractals that are probably the most liked part of my work.
I would dearly love to see the pentation version of this tetration fractal. Thirty years of more math knowledge, better programming tools, much greater computer speed and memory. I'll be seeing what can be done with Julia to this end.
Daniel
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07/20/2022, 07:41 PM
(This post was last modified: 07/20/2022, 07:45 PM by bo198214.)
Also thank you from my side, MphLee. You are a great mediator.
To Daniel: Though it sometimes seems different, I am glad to have you on the forum. I told you what point sets me up, and maybe a bit of quarrel is than unavoidable, but still you are welcome here. And after all we Tetration folks should stick together.
(07/19/2022, 10:52 PM)Daniel Wrote: I would dearly love to see the pentation version of this tetration fractal. Thirty years of more math knowledge, better programming tools, much greater computer speed and memory. I'll be seeing what can be done with Julia to this end.
Should be doable already with Sheldon's or Paulsen's method.
This is also a Tetration Fractal, who is it?
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(07/20/2022, 07:41 PM)bo198214 Wrote: Also thank you from my side, MphLee. You are a great mediator.
To Daniel: Though it sometimes seems different, I am glad to have you on the forum. I told you what point sets me up, and maybe a bit of quarrel is than unavoidable, but still you are welcome here. And after all we Tetration folks should stick together.
(07/19/2022, 10:52 PM)Daniel Wrote: I would dearly love to see the pentation version of this tetration fractal. Thirty years of more math knowledge, better programming tools, much greater computer speed and memory. I'll be seeing what can be done with Julia to this end.
Should be doable already with Sheldon's or Paulsen's method.
This is also a Tetration Fractal, who is it?
Thank you for your kind words Bo. I accept what you have said as truth and therefore no longer have any issue with you. Consider me "annealed" as I no longer have a problem with being a member of the Tetration Forum community and withdraw my request for my account to be deleted.  Cheers
Daniel
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(07/20/2022, 07:41 PM)bo198214 Wrote: This is also a Tetration Fractal, who is it?
:)))) Baphomet?
Gottfried Helms, Kassel
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Oh I meant nothing against you Daniel.
Please don't feel personal slight against you. But it seems sometimes you think the only tetration is your tetration, and it can be frustrating. Where as mathematicians, we should be able to identify there are uncountably infinitely many tetrations. And this doesn't even include the smooth tetrations.
There is so much to iteration theory beyond the Bell polynomial approach. But you seem to write these off. It doesn't really bother me, until it appears you deny its existence. I have nothing but respect for you. If anything, I think you take criticisms a tad too personal.
I apologize if I went off and spoke badly towards you. That wasn't my intention. But I come from a family of stern hands and stern words. Smarten up and stand straight is a common tongue I know far too well.
I like having you here, and I like your questions, and I like your posts. I fucking love your fractals, and I respect your approach at mathematics. There is nothing there that I don't. I guess it seems that sometimes you don't understand what a Schroder function is, and therefore it doesn't exist. And that's very frustrating. I can, and just as well bo can, construct the bell polynomial approach, and unfortunately it uses mathematics that's been around for 50  100 years depending on who you reference. It can be frustrating from my perspective, when you deny that it's Schroder's method, when I can guarantee it is. And as mathematicians, ignorance is not an excuse. Maybe in the social sciences (oh I didn't read that article) that defense will work. Not in mathematics.
Again, nothing but respect for you, Daniel. I do not think you act like you're a sage, bo said that. I said holier than thou, and I apologize. I get now that may have been a tad disrespectful.But remember, in the wise words of my grade 11 math teacher who used to quote famous mathematicians from their personal letters.
Quote:You idiot! You don't even know what a hyperbola is!  Descartes, to some other mathematician
These heated debates have been going on for ever. Let's change the tune. And to do that, we need to free ourselves from an emotional attachment to the mathematical object. The bell polynomials are fantastic, your work is very well donebut there are many upon many ways of constructing the same construction.
I hope I am not being aggressive or mean, that can come off from my language. I honestly don't have the energy for drama on the tetration forum. I get enough of that from the women in my life
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07/21/2022, 02:19 AM
(This post was last modified: 07/21/2022, 02:21 AM by Daniel.)
(07/20/2022, 11:38 PM)JmsNxn Wrote: Oh I meant nothing against you Daniel.
Please don't feel personal slight against you. But it seems sometimes you think the only tetration is your tetration, and it can be frustrating. Where as mathematicians, we should be able to identify there are uncountably infinitely many tetrations. And this doesn't even include the smooth tetrations.
There is so much to iteration theory beyond the Bell polynomial approach. But you seem to write these off. It doesn't really bother me, until it appears you deny its existence. I have nothing but respect for you. If anything, I think you take criticisms a tad too personal.
I apologize if I went off and spoke badly towards you. That wasn't my intention. But I come from a family of stern hands and stern words. Smarten up and stand straight is a common tongue I know far too well.
I like having you here, and I like your questions, and I like your posts. I fucking love your fractals, and I respect your approach at mathematics. There is nothing there that I don't. I guess it seems that sometimes you don't understand what a Schroder function is, and therefore it doesn't exist. And that's very frustrating. I can, and just as well bo can, construct the bell polynomial approach, and unfortunately it uses mathematics that's been around for 50  100 years depending on who you reference. It can be frustrating from my perspective, when you deny that it's Schroder's method, when I can guarantee it is. And as mathematicians, ignorance is not an excuse. Maybe in the social sciences (oh I didn't read that article) that defense will work. Not in mathematics.
Again, nothing but respect for you, Daniel. I do not think you act like you're a sage, bo said that. I said holier than thou, and I apologize. I get now that may have been a tad disrespectful.But remember, in the wise words of my grade 11 math teacher who used to quote famous mathematicians from their personal letters.
Quote:You idiot! You don't even know what a hyperbola is!  Descartes, to some other mathematician
These heated debates have been going on for ever. Let's change the tune. And to do that, we need to free ourselves from an emotional attachment to the mathematical object. The bell polynomials are fantastic, your work is very well donebut there are many upon many ways of constructing the same construction.
I hope I am not being aggressive or mean, that can come off from my language. I honestly don't have the energy for drama on the tetration forum. I get enough of that from the women in my life
Hey JmsNxn, it's all good.
You raise several issues or concerns. You believe I think that my work is the only valid approach to extending tetration. I have no respect for the advances of iteration theory. I don't understand Schroeder's function and that my work is basically a warmed over version of Schroeder's function.
My presence in the community and my offer to help preserve the Tetration Forum should be proof that I value the knowledge here. My limitations in iteration theory have more to due with my inability to read German or French. I'm very appreciative of Bo's writeup of Kneser's work and even added a link to it on Tetration.org's main page. I tried to read about Ecalle' work and got my butt kicked on the first page. I feel that to really understand fractional iteration I need to work in historical order through all the central papers on iteration. Considering my age and that technically I don't sleep, I don't think that is going to happen. I did try to read Trappmann and Kouznetsov's work without success. But that doesn't mean I shouldn't try again. This is why I was asking questions about what the main approaches to extending tetration are. I even said I suspected we might be looking at different parts of the same elephant. Paulsen's work is consistent with Kouznetsov's; my work and Gottfried's also appear to be consistent and quite possibly related. I would be happy is all the different techniques gave valid extensions to tetration. Then we might be able to mix and match theorems from different techniques and maybe find an even greater overarching theory. Maybe I could adapt Paulsen's work on convergence with my own work.
Why do you believe you understand the scope of my work so well you can dismiss it in a sentence? Why do you believe you have knowledge of what goes on in my mind? What portion of my work have you read? Just the postings here? Do you understand the material on my website and most importantly have you read a major summation of my work, Bell Polynomials of Iterated Functions?
My earliest work on iteration was basically equivalent to Schroeder's function. That much is true. Then I realized that my equation blew up when the multiplier was 1. So then I worked with the fractional iteration of functions multiplier was 1 like sin and exp(x)1. My results in with exp(x)1 are identical with Gottfried's work with matrices.
Then I had a cool thought. All my work was based on summations of geometrical progressions. Well I applied the basic formula for the summation of geometrical progressions, but the formula blows up at 1. So my equation was bound to blow up when the multiplier was 1. By postponing the evaluation of the summations, my equation was now able to handle both Schroeder and Abel's equations. More than that, it allowed both the properties of Schroeder and Abel's equations to be derived. And while my work appears simple, that is something I strive for. It allowed for the combinatorial structure to be identified. Just like Feynman diagrams, each combinatorial structure enumerated and then evaluated and summed together. So I can think of iterated functions in terms of Schroeder's Fourth Problem, also know as hierarchies, phylogenetic trees, and total partitions. Basically it is a rooted tree that is the recursive version of set partitions. Yup, same Schroeder. Our pal published this a year before his famous paper on iterated functions. So my work is a synthesis of Schroeder and Abel's equations, and explanation of the classification of fixed points and a combinatorial interpretation of iterated functions.
Feel free to challenge me on any of this, although hopefully not everything as I'm tying to enjoy my retirement.
Daniel
Posts: 1,214
Threads: 126
Joined: Dec 2010
(07/21/2022, 02:19 AM)Daniel Wrote: (07/20/2022, 11:38 PM)JmsNxn Wrote: Oh I meant nothing against you Daniel.
Please don't feel personal slight against you. But it seems sometimes you think the only tetration is your tetration, and it can be frustrating. Where as mathematicians, we should be able to identify there are uncountably infinitely many tetrations. And this doesn't even include the smooth tetrations.
There is so much to iteration theory beyond the Bell polynomial approach. But you seem to write these off. It doesn't really bother me, until it appears you deny its existence. I have nothing but respect for you. If anything, I think you take criticisms a tad too personal.
I apologize if I went off and spoke badly towards you. That wasn't my intention. But I come from a family of stern hands and stern words. Smarten up and stand straight is a common tongue I know far too well.
I like having you here, and I like your questions, and I like your posts. I fucking love your fractals, and I respect your approach at mathematics. There is nothing there that I don't. I guess it seems that sometimes you don't understand what a Schroder function is, and therefore it doesn't exist. And that's very frustrating. I can, and just as well bo can, construct the bell polynomial approach, and unfortunately it uses mathematics that's been around for 50  100 years depending on who you reference. It can be frustrating from my perspective, when you deny that it's Schroder's method, when I can guarantee it is. And as mathematicians, ignorance is not an excuse. Maybe in the social sciences (oh I didn't read that article) that defense will work. Not in mathematics.
Again, nothing but respect for you, Daniel. I do not think you act like you're a sage, bo said that. I said holier than thou, and I apologize. I get now that may have been a tad disrespectful.But remember, in the wise words of my grade 11 math teacher who used to quote famous mathematicians from their personal letters.
Quote:You idiot! You don't even know what a hyperbola is!  Descartes, to some other mathematician
These heated debates have been going on for ever. Let's change the tune. And to do that, we need to free ourselves from an emotional attachment to the mathematical object. The bell polynomials are fantastic, your work is very well donebut there are many upon many ways of constructing the same construction.
I hope I am not being aggressive or mean, that can come off from my language. I honestly don't have the energy for drama on the tetration forum. I get enough of that from the women in my life
Hey JmsNxn, it's all good.
You raise several issues or concerns. You believe I think that my work is the only valid approach to extending tetration. I have no respect for the advances of iteration theory. I don't understand Schroeder's function and that my work is basically a warmed over version of Schroeder's function.
My presence in the community and my offer to help preserve the Tetration Forum should be proof that I value the knowledge here. My limitations in iteration theory have more to due with my inability to read German or French. I'm very appreciative of Bo's writeup of Kneser's work and even added a link to it on Tetration.org's main page. I tried to read about Ecalle' work and got my butt kicked on the first page. I feel that to really understand fractional iteration I need to work in historical order through all the central papers on iteration. Considering my age and that technically I don't sleep, I don't think that is going to happen. I did try to read Trappmann and Kouznetsov's work without success. But that doesn't mean I shouldn't try again. This is why I was asking questions about what the main approaches to extending tetration are. I even said I suspected we might be looking at different parts of the same elephant. Paulsen's work is consistent with Kouznetsov's; my work and Gottfried's also appear to be consistent and quite possibly related. I would be happy is all the different techniques gave valid extensions to tetration. Then we might be able to mix and match theorems from different techniques and maybe find an even greater overarching theory. Maybe I could adapt Paulsen's work on convergence with my own work.
Why do you believe you understand the scope of my work so well you can dismiss it in a sentence? Why do you believe you have knowledge of what goes on in my mind? What portion of my work have you read? Just the postings here? Do you understand the material on my website and most importantly have you read a major summation of my work, Bell Polynomials of Iterated Functions?
My earliest work on iteration was basically equivalent to Schroeder's function. That much is true. Then I realized that my equation blew up when the multiplier was 1. So then I worked with the fractional iteration of functions multiplier was 1 like sin and exp(x)1. My results in with exp(x)1 are identical with Gottfried's work with matrices.
Then I had a cool thought. All my work was based on summations of geometrical progressions. Well I applied the basic formula for the summation of geometrical progressions, but the formula blows up at 1. So my equation was bound to blow up when the multiplier was 1. By postponing the evaluation of the summations, my equation was now able to handle both Schroeder and Abel's equations. More than that, it allowed both the properties of Schroeder and Abel's equations to be derived. And while my work appears simple, that is something I strive for. It allowed for the combinatorial structure to be identified. Just like Feynman diagrams, each combinatorial structure enumerated and then evaluated and summed together. So I can think of iterated functions in terms of Schroeder's Fourth Problem, also know as hierarchies, phylogenetic trees, and total partitions. Basically it is a rooted tree that is the recursive version of set partitions. Yup, same Schroeder. Our pal published this a year before his famous paper on iterated functions. So my work is a synthesis of Schroeder and Abel's equations, and explanation of the classification of fixed points and a combinatorial interpretation of iterated functions.
Feel free to challenge me on any of this, although hopefully not everything as I'm tying to enjoy my retirement.
I do not think your work is simple. I have seen your website. I have read a couple of your papers. As I said it's fantastic. I do not think your work is simple. I do not think many of the things you said that I said. I'm all for your work Daniel. We are all in this together.
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I've reread some of my posts and folks were right, I've been on an ego trip. Something can be factual without being truth. Any portrayal of me as a significant person is at best only a half truth. So folks have my apology.
Peace,
Daniel
Posts: 1,214
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Joined: Dec 2010
There's no need to apologize, Daniel. We're all colleagues here, if there is such a thing as internet colleagues, lol. Just remember we're all working towards a common goal. We need to solve tetration! and that means we need to describe every part of the elephant!
