Applications of Tetration - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Applications of Tetration (/showthread.php?tid=105) Pages: 1 2 Applications of Tetration - andydude - 01/01/2008 I just wanted to show off a paper. I wrote this for my English class, because my teacher required we write a paper in the subject of our major, and since my major is math, I decided to do something that interests me. I left a few applications out due to the length of the paper with them included, some of the applications I know about, but either omitted or I wish I could talk more about: Applications of the super-logarithm to population modeling. Applications of the Lambert W function $(W(x) = {}^{\infty}(e^{-x})x)$ related to tetration. Hierarchies of height m on n nodes (trees with labeled leaves) $(DE_e^m(x))^{(n)}(0)$ Forests of height less than m on n nodes (trees with labeled nodes and unlabeled root) $(x \rightarrow {}^m{(e^x)})^{(n)}(0)$ Trees of height m on n nodes (trees with labeled nodes and labeled root) $(x \rightarrow {}^m{(e^x)}x)^{(n)}(0)$ Who knows what this generates! OIES has no applications. $(x \rightarrow {}^{m}x)^{(n)}(1)$ Fictional animal chase (Devaney's "Playing catchup with iterated exponentials") Fictional corporate policy (a story I wrote) Any spelling errors, corrections, comments, would be most welcome. Here's the paper: Applications of Iterated Exponentials. Andrew Robbins PS. Happy New Year! RE: Applications of Tetration - Ivars - 01/02/2008 Absolutely great! But allow me a comment as a conclusion of what I read: - mathematics is not finished without analytical theory on tetration, especially infinite since it is natural completion of infinite sums and products, and basically should not be more complex than other operations -since mathematics is not finished no physical theory which does not imply usage of infinite tetration can be considered finished, especially in part where Andrew talks about rotation through dimensions. Nature is much more complex than addition and multiplication and simple exponents. -the problem of analytical theory of tetration can not be solved without stepping outside real number line, limits and understanding the scale dependency of imaginary unit i. An example: We say h(e^pi/2) = i; -i is not tetration of e^pi/2, since real number > e^1/e infinitely tetrated can not become complex number as a limit of convergent series of real numbers. Also we say that limit must have 1 value. Full stop. Why don't we consider that since h(s) where s> e^(1/e) are all complex, and multiple, that it is tetration, but something is wrong with real number line and definition of limit as applied to tetration? It is obvious that tetration is an operation that WANTS to push real numbers into complex plane, but this obvious fact is dismissed as garbage since holy real number line as it is constructed today would not allow it. Tetration is yelling at us that something is wrong with basic numbers since they do not accomodate such a natural thing as tetration, and thus in 230 years since Euler no big advances has been made. And he did not care much about rigorous use of either limits or sets. So why not change it-discard and rebuild - the real number line , complex numbers so that they fit tetration? That would be a million dollar worth achievement, and solve all other prized problems on a way. Hyperreals fall short. Scaled infinitesimals and infinities do not - but that is my personal opinion. I have not even heard about such a suggestion, so probably there is nothing to read about it either- which just offers more opportunities for people who really understand tetration-like You ( not me). Happy New Year! RE: Applications of Tetration - andydude - 01/03/2008 I would like to address each comment you made: Ivars Wrote:mathematics is not finished ... Mathematics is never finished! Goedel's incompleteness theorem even proves it! Ivars Wrote:... is not finished, no ... which does not ... "Simply stated, it is sagacious to eschew obfuscation." -- Norman R. Augustine Ivars Wrote:analytical theory of tetration can not be solved ... The analytical theory of tetration is coming along great with the real number line as is, but requires complex numbers for some bases. There is nothing wrong with the real number line, period. There are many things that make other systems much nicer than the reals, for example algebraic closure of the complex numbers, the transfer principle of the hyperreal numbers, etc. Also, there are many cases where definitions are extended beyond their original domains, for example $\sum_{k=1}^{n}x^k = \frac{1}{1-x}$ but the series was only defined for $-1 < x < 1$. This does not mean the definition was wrong, it just means the latter is a formula with a larger domain. I recognise that $h_{(-1)}\left(e^{\pi/2}\right) = i$ is evidence that tetration pushes real numbers into the complex plane, I accept that. You need to learn to keep your cool. Ivars Wrote:... rebuild the real number line, complex numbers so that they fit tetration? If you read Henryk Trappmann's paper on binary tree arithmetic available from the top of this web site, that is pretty much what he does. He effectively re-builds the real numbers using binary trees (as far as I understand) and he does this to better fit tetration. You are not alone. Andrew Robbins RE: Applications of Tetration - Ivars - 01/03/2008 Hi Andrew, Many thanks for serious answers and link. I hope You were not too annoyed with my style.. Quote:[*]"Simply stated, it is sagacious to eschew obfuscation." -- Norman R. Augustine I agree, but is not it wonderful how easily physicists would dismiss such a basic operation. Or where they mathematicians? So I am looking for a reason that happened, and, if the reason is sticking to certain rigour than that would be the first place I would be looking for fundamental mistake regardless of achievements and theories that are based on that rigour. 80/20 principle usually works better than gradual approach. Quote: I recognise that $h_{(-1)}\left(e^{\pi/2}\right) = i$ is evidence that tetration pushes real numbers into the complex plane, I accept that. You need to learn to keep your cool. Very happy to hear that, as well as Your positive opinion about series summation via 1/1-x. Best regards, Ivars RE: Applications of Tetration - Ivars - 01/12/2008 Hej Andydude Here is one application - a little bit far fetched- but I just composed it today- from physics of consciousness - if that is a true conjecture, than many things in the middle will require attention: Please look at this article:"Dimensions of consciousness" http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1201004 Though I do not know exactly how fractal dimensions of phase space are obtained, so I have to be careful with interpretations,may be it is pure coincidence, but numerically they might make sense, especially this table from the article: Table Rankings of highest fractal dimensions obtained from the electroencephalographs of 11 species SpeciesHighest fractal dimension Human4.85 Dog4.63 Bullfrog3.71 Minnow3.09 Catfish2.50 Perch2.37 Crayfish1.65 Earthworm0 Moth larva0 Starfish0 Anemone0 If You look at value e^pi/2=4,810................ The coincidence is quite interesting. One would say that dimensionality e^pi/2 would be ideal, as it would mean perfect value h(e^pi/2) = - i . Could it be that this fractal dimension shows the result of starting with e.g -i and taking infinite roots of it - kind of opposite operation of infinite exponetiation?if we start with - i and If You achieve 4,81...- perfect. 4,85 is a slight overshoot. The same happens, if one start with i - only pure infinite tetration backwards leads to e^pi/2. Opposite conjugate values, the same consciousness- 2 sexes? Or if we start with 4,85 and perform infinite tetration, the result will have both imaginary close to -i component and a small Real one. In infinite tetration h(z) , if z>=4,85> 4,81...=e^pi/2, real part of h(z) becomes negative. So there is a clear phase transition division line between humans and the rest. If z<4,81 ( dogs and the rest), real part of h(z) is POSITIVE, and becomes bigger while imaginary gets bigger as well as z gets smaller. I have attached a curves by Gottfried Helms /Andydude where it is clearly visible. http://math.eretrandre.org/tetrationforum/attachment.php?aid=78 http://tetration.itgo.com/up/selfroot-real-zeros.png http://go.helms-net.de/math/tetdocs/RealvaluedYtraces4_emboss.png As z gets smaller, we move to the right along the curve- at z =b=3 (Minnow) Real part of h(3) is already +0,24. Interestingly, due to properties of infinite tetration, it converges ( h(b) imaginary part becomes 0) when b< e^(1/e) = 1,444668. So there is another phase transition- infinite tetration of numbers below that does not lead to emergence of imaginary component- so no consciousness. Crayfish has b=1,65, and after that everything is 0. No consciousness. Other interesting conclusion from this exercise could be the meaning of numbers. Essentially, these numbers represent some dimensionality in phase space of some complex non-linear process - and not at all simple points on the real line. I think the number of meaningful variables/dimensions involved is much bigger, since to reach i from 4,81 ... You need to perform infinite number of infinitesimal operations. Here in this forum Gottfrieds Im(s)=0 for s>e^(1/e) and its variations are well known, so obviously there is much more in it, but at main value level things seem reasonable. Weaving the Web of life, I would call the full graph with branches, as it resembles a spider. How could it be easily checked if there exist a relation between tetration and chaos dynamics, fractal dimensions authors mention in the text? Most likely there is connection, since tetration anyway is a missing operation, but probably not well known, these connections are. Best regards, Ivars Fabriciuss RE: Applications of Tetration - bo198214 - 01/19/2008 @Andrew This is a very nice text and I wonder whether it should be incorporated into the FAQ. Because surely one frequently asked question is "what are the applications of tetration?". @Ivars That sounds quite interesting, however it is perhaps more a philosphical and speculative approach, which may perhaps not find that many partisans at this forum. RE: Applications of Tetration - Ivars - 01/20/2008 bo198214 Wrote:@Ivars That sounds quite interesting, however it is perhaps more a philosphical and speculative approach, which may perhaps not find that many partisans at this forum. hej bo I noticed that. However, I sent it also to the authors and we are trying to find a link between fractal dimensions in chaos dynamics and infinite tetration, since they have proceeded measuring return of cosnciousness and found interestind phase transitions between types of attractors at dimensions approx 1,5 and 2,5. The idea that tetration is a process that allows to "visit future" -escape in imaginary time- to check where the process should go by going into imaginary time while real time stops has been also applied to tornado emergence simulations in 1970-ties. It is not much, but the idea of tetration sitting inside phase transitions just because of its immense speed of growth and escape to complex values is one worth considering. Is this forum interested in finding links between tetration and so called "chaos" dynamics? Ivars RE: Applications of Tetration - andydude - 01/20/2008 bo198214 Wrote:This is a very nice text and I wonder whether it should be incorporated into the FAQ. Because surely one frequently asked question is "what are the applications of tetration?". Sadly, its the only question I ever get asked. PS. (this is referring to Real Life, not this forum...) RE: Applications of Tetration - andydude - 01/20/2008 @Ivars: First, the process of turning brainwaves into voltages is not an exact science. This means that the voltages that are accessible via an EEG are not going to be exactly what is going on in your brain (although its pretty close). Second, the process of turning raw data into fractal dimensions is not an exact science. Wikipedia even has a warning about this. The reason why is because fractal dimensions were originally defined for objects with self-similarity, and mathematically-defined patterns. Raw data does not have this kind of self-similarity, and if it does, then it is enforced upon the raw data, and not something that is somehow encoded within (or perhaps I am enforcing randomness on all data). Aside from these remarks, it would be an interesting discovery, that: tetration explains our brains. Andrew Robbins RE: Applications of Tetration - Gottfried - 01/20/2008 andydude Wrote:bo198214 Wrote:This is a very nice text and I wonder whether it should be incorporated into the FAQ. Because surely one frequently asked question is "what are the applications of tetration?". Sadly, its the only question I ever get asked. PS. (this is referring to Real Life, not this forum...) ... not surprising: first consider the relevance of a theory, then develop it into detail... (This does not apply to the "musicians" in number-theory, though ) Gottfried