Talk:Tetration/Archive 2

Ultra exponential
I see no objection to the merge. Any further comments? — Arthur Rubin (talk) 07:37, 8 October 2009 (UTC)


 * I've merged, let me know what you guys think. There's a lot more work that could be done to fit it in with the rest of the section on extending tetration to real heights. Cheers, — sligocki (talk) 08:11, 17 October 2009 (UTC)

For me it is not obvious, why the sequence of more-and-more generalizations, for instance "extension to negative height","infinite height", "extension to real height","extension to complex height" should be disrupted by the subject "polynomial " and "ultraexponential". (For me it looks now also similar with the sequence of generalizations concerning the base-parameter). Actually just these extensions can be used as milestones if once a history of the extension of the sequence of operators to a fully recognized and applicable method of iterated exponentiation can be presented. Gottfried --Gotti 10:53, 19 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

a^a, (b-1) times?
It seems to me that the expression ba should be equal a raised to the a power, b-1 times. This is because the article states, for example, that 23 is actually 33, which is raised to the 3rd power not twice (b=2), but only once. Am I just getting the wrong concept to this? (sorry for the lack of fancy math script, I'm not extremely familiar with the syntax) -Kanogul (talk) 03:13, 8 October 2009 (UTC)


 * It is a loose formulation, "chained power involving b numbers a", a little lower, is more accurate.--Patrick (talk) 09:52, 8 October 2009 (UTC)


 * OK. I understand, it was just kind of bugging me.  Thanks for clearing that up. -Kanogul (talk) 04:06, 9 November 2009 (UTC)

addition is not the bottom
The bottom is x+1. F(x)= x+1

Addition +(x,y) is F(x) y times, or in my made-up notation (y)F(x,whatEverYDoesNotMatterInThisFunction).

Multiplication is 0 + x y times x(x,y)=(y)+(0,x)

Exponentiation (maybe backwards exponentiation, or raising)is 1 multiplied by x y times ^(x,y)=(y)x(1,x)

Tetration (actually backward tetration) is x root of x raised by x y times M(x,y)=(y)^(xrootx,x)

If f(x,y)=z, ||f(y,x)=z, ~f(z,y)=x

Tetration would be M2(x,y)=(y)||^(1,x)

So the most fundamental of these operations is x+1, as is explained on the hyperoperation page.

@(f,n)(x,y) = (n)f(x,y)

(0)f(x,y)=x

(-1)f(x,y)=~f(x,y)

(2)f(x,y)=x o f(x,y) o f(x,y)=f(x,y) o f(x,y) =f(f(x,y),y)

(m)f((n)f(x,y),y)=(n+m)f(x,y)

(-n)f(x,y)=(n)~f(x,y)

f->n(x,y)=(y)f(n,x)

—Preceding unsigned comment added by 72.66.99.191 (talk) 01:45, 16 March 2010 (UTC)

update of a web-link impossible
For some monthes now the homepage of A.Robbins has moved from itgo.com to tetration.co.cc. It was impossible to adapt that links in the reference- and weblinks-sections ("spamschutz"). The page of Robbins contains serious and valuable material, so this seems unfortunate. On the other hand I agree to see "spamschutz" here in general. So maybe some editor/caretaker is willing and able to look at this and update that links.

Gottfried Helms —Preceding unsigned comment added by Druseltal2005 (talk • contribs) 14:00, 30 September 2010 (UTC)

--Gotti 13:57, 30 September 2010 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

7th century documents
The source seems to be self-published, even if it might say what is quoted. I've reverted, but the analysis of that notation needs to be in a reliable source. — Arthur Rubin (talk) 20:13, 1 October 2010 (UTC)

Why reject the link ?
I tried to plug in the relevant links : http://sites.google.com/site/tommy1729/tetration

and http://sites.google.com/site/tommy1729/extended-distributive-property

But my edit got cancelled.

There is no mistake in those links and i did not remove anything.

Therefore i feel insulted and i object. —Preceding unsigned comment added by 81.240.130.175 (talk) 20:26, 9 January 2011 (UTC)


 * It's a personal website by a person with no indication of expertise or accuracy, and it introduces new notation. If someone else used the notation, than even a personal website might be considered an acceptable external link.  — Arthur Rubin  (talk) 20:41, 9 January 2011 (UTC)
 * In addition, the first link "tetration" is just wrong. That doesn't work at all.  — Arthur Rubin  (talk) 20:44, 9 January 2011 (UTC)

that is nonsense, the person is an expert and a regular respected poster on the " tetration forum " , what he said is correct and his notations are consistant.

look at the tetration forum if you dont believe it.

and it isnt just wrong, it seems you dont get it.

well you dont get it, i cant say that more politely ... —Preceding unsigned comment added by 81.240.67.178 (talk) 23:33, 10 January 2011 (UTC)


 * It's highly improbable for real x, and completely wrong for complex z, as the approximation $$e^z \approx 2 \sinh z$$ is only valid for z having large real positive real part, and that is not preserved as k goes to infinity. Furthermore, even if the resulting function exists, it almost certainly fails regularity conditions, and would produce  different functions if one replaced $$2 \sinh x\,$$ by $$2 e^{-a} \sinh (x+a)\,$$, for any real a.
 * And, for what it's worth, I worked on the functional square root of the exponential function long before the term tetration was coined. — Arthur Rubin  (talk) 02:39, 11 January 2011 (UTC)
 * And, also, for what it's worth, I am a recognized expert on weak forms of the axiom of choice, and the existence of non-measurable sets is a clear consequence of that. — Arthur Rubin  (talk) 02:49, 11 January 2011 (UTC)


 * Returning to the distributive law, that applies to balanced hyperoperations, and might conceivably be an appropriate external link in hyperoperations, if it's the same balanced hyperoperations as we're using. — Arthur Rubin  (talk) 09:14, 11 January 2011 (UTC)

sigh, its not highly improbable for real x , its TRUE for real x.

also it requires a trick to extend it to complex z, which you would know if you read it carefully , understand it and are honest about it.

and yes it is preserved as "k" goes to infinity at least for real x.

$$2 \sinh x\,$$ by $$2 e^{-a} \sinh (x+a)\,$$, for any real a are just two different functions, it doesnt make sense as an argument.

the axiom of choice has nothing to do with a page about tetration.

none of your counterarguments made sense, let alone a disproof. saying it doesnt work doesnt mean it doesnt work. it does work.

furthermore if you look at the tetrationforum you will see it is taken very seriously.

most other members of that forum have made a link to their site on this tetration page, hence combining all that , i deserve one too.

if you are the famous Rudin, i advice you to reconsider your opinion even more for the sake of your reputation.

because make no mistake, i have more up my sleeve than a free google website. —Preceding unsigned comment added by 81.240.67.178 (talk) 19:37, 11 January 2011 (UTC)


 * At wikipedia we avoid personal attacks. If you continue in this vein your account will probably be blocked.  The test of whether the results you cite are correct is whether they are published in a peer-reviewed periodical, not whether a wikieditor is absolutely convinced of their correctness.  Tkuvho (talk) 20:18, 11 January 2011 (UTC)

you talk about peer review.

but there are lots of things on wiki without peer-review.

the website links by the others : did they have peer-review ? No.

are they relevent ? YES.

tetration is a controversial subject, and they dont get into journals easily.

but tommy's result clearly deserves a link as much as the others.

links are never peer reviewed anyways ?

only kouznetsov got published.

and kneser, but he is not even in the article.

ive never seen proof a peer-review of all links in a wiki article ! —Preceding unsigned comment added by 81.240.67.178 (talk) 22:26, 11 January 2011 (UTC)


 * Very little in this field is published, and I might be willing to accept non-published papers if not obviously wrong. In these specific examples:
 * My reference to the axiom of choice was to another note on http://sites.google.com/site/tommy1729, on a topic on which I am a recognized expert, and noting that it is completely and totally wrong. That doesn't mean he's wrong about tetration, but it does cast doubt on his expertise.
 * About the "tetration" article:
 * I would like to see a proof that the sequence converges, and that it satisfies the functional equation. If that would exist, I would probably accept it as an external link, even if it were totally wrong for complex numbers.  There is also no reason to believe that the resultant complex function is even real-analytic, and it's not even obvious that it's infinitely differentiable.  If that's not the case, there's no "Taylor series" or "analytic continuation" to be done.
 * The choice of asymptotic approximation to ex should affect the resultant functional square root, making it not produce a "canonical" function. It would still be of interest if correct
 * About the "distributive" article:
 * It doesn't seem to apply to the standard forms of tetration or any hyperoperation; it's closer to the balanced hyperoperations, but not quite there. I'd rate it as non-notable.  — Arthur Rubin  (talk) 00:34, 12 January 2011 (UTC)
 * And, if "you" (81.240.*.*) are tommy1729, you shouldn't be posting this, per WP:COI. I may be too sensitive to that, but I haven't added my parents' books as references on the axiom of choice, even though they are recognized as some of the best collections.  — Arthur Rubin  (talk) 01:04, 12 January 2011 (UTC)
 * I do not think that using the books would be a big problem. For one, COI is a guideline, not a policy. For another, if those books rae academic high-quality sources as you say, then they definitely would be suitable for WP, you just need to hold to content policy when using them, just like any other source. If some subtle biases creep in, this could be fixed. If you are that worried about it you could always ask on the relevant talk pages when you put in the edits. mike4ty4 (talk) 03:44, 20 February 2011 (UTC)
 * And by "he shouldn't post" do you mean "post links" or "post talk"? The former is what the guideline pushes against, the latter is not covered. mike4ty4 (talk) 03:44, 20 February 2011 (UTC)

Something intresting about Arthur Rubin :

http://math.eretrandre.org/tetrationforum/showthread.php?tid=312&pid=5518#pid5518

post number 1 till 8.

seems you are playing games and are now " unmasked ".

since you are on the tetration forum, you must have seen the proofs posted that you ask for here.

more particular

http://math.eretrandre.org/tetrationforum/showthread.php?tid=424

post number 9 :

but it is proven to be infinitely differentiable in [2].

[1] Lévy, P. (1927). Sur l'itération de la fonction exponentielle. C. R., 184, 500–502. [2] Walker, P. L. (1991). Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput., 57(196), 723–733

post number 38 :

Ok, first let us verify that it is indeed an iteration of exp, i.e that indeed satisfies: and.

neglecting some rules of properly evaluating limits we get .

and because towards infinity gets arbitrarily close to.

btw : if you cant see the math symbols go to that post number 38.

furthermore those proofs are by people other than tommy himself, in case you cant check the math which is perfectly correct.

also intresting

post number 12 :

As might be expected, Tommy's sexp gives different values. The function also has different values than the base change solution, discussed earlier in this forum, which Bo points out is equivalent to Lévy's solution. So, it seems we have yet another super exponential, different from all the others. - Sheldon

convinced ? or will you continue playing games ? —Preceding unsigned comment added by 81.240.238.139 (talk) 21:23, 15 January 2011 (UTC)


 * I read the forum thread after I questioned the accuracy, and my comments mostly stand.
 * The critical phrase is "neglecting some rules of properly evaluating limits". No one on the tetration forum has provided evidence that that can be done safely.  Furthermore, according to other comments in the thread (also without proof), Tommy's solution is infinitely differentiable and nowhere analytic, making his "extension to the complex plane" statements disproved, rather than merely unproved.
 * I'm not entirely sure that Tommy's method (with $$e^x-e^{-x}$$) is the same as Levy's method (with $$e^x-1$$), but, if it were, then Levy's paper is a better source, and, Tommy's web site adds little (except for the believed analytic continuation). Also, there would be a continuous-dimensional family of solutions, for any analytic approximation to the exponential function which has exactly one real (repelling) fixed point.  (Levy's is parabolic repelling, and Tommy's is hyperbolic, and that may or may not be a significant difference.)  — Arthur Rubin  (talk) 23:21, 15 January 2011 (UTC)
 * More original research on my part, suggesting that Tommy's method might be significantly worse than Levy's should be left to more speculative forums. — Arthur Rubin  (talk) 23:35, 15 January 2011 (UTC)

How dare you ??

you asked for proof and evidence, i provided it.

you brake your promise !

as for the " neglecting some rules ", it had a smiley behind it ; it was said jokingly !

furthermore i know it isnt analytic by that limit, it is " realanalytic "

you are right about " im not entirely sure ", in fact you know nothing about it.

quoting you : " I would like to see a proof that the sequence converges, and that it satisfies the functional equation. If that would exist, I would probably accept it as an external link, even if it were totally wrong for complex numbers. "

i provided you that !

so you LIED !

you are a dishonest person and should be ashamed of yourself ! —Preceding unsigned comment added by 81.240.96.134 (talk) 21:30, 16 January 2011 (UTC)


 * There's no proof or claim of a proof on the tetration forum, at least not one in or pointed to by that thread.
 * There's a claim in the thread that it's not real-analytic, although (at least Levy's construction) is infinitely differentiable.
 * And none of this is relevant. Until there's a published proof, or a claim that it's better than Levy's construction, it doesn't belong in Wikipedia, whether or not correct.  We prefer published papers to web sites which are clearly not reliable sources.  — Arthur Rubin  (talk) 06:16, 17 January 2011 (UTC)

the other links are not " published " either ! —Preceding unsigned comment added by 81.240.96.134 (talk) 21:53, 17 January 2011 (UTC)


 * We can use unpublished (and unreliable) works to determine the credibility of other unpublished works, or we cannot include them at all. Tommy1729 does not appear to qualify as an expert per WP:SPS.  — Arthur Rubin  (talk) 22:29, 17 January 2011 (UTC)

Why don't you do the operator ' binary? (example: a'a'a=a+3 ) 192.116.88.204 (talk) 00:32, 12 March 2011 (UTC)

In the paragraph "Iterated powers" you show how the higher operators get more complex because the Importance of right order of operators. The bigger number of times of exponentiations of A to itself, The bigger number of options to order the operators. See Catalan Numbers. 192.116.88.204 (talk) 00:52, 12 March 2011 (UTC)

more theoretical operators
What about a sign for:  a÷a÷...a  n times =a#n

What about a sign for:  a$a$...a  n times =a÷n

or a"a"...a  n times =n÷a

Sometimes we need brackets because the order of the operators is not always right to left. Can it happen also in calculate of two numbers? like n-a that supposed to be a-n ...Is it understandable?

“ a added to itself, n times. ”--Make sure that it's understandable enugh that "a added to itself, one time" does not mean a+a. In this exercise there is 1 time +. It has 1 a, 1 added, and 1 "itself". 192.116.88.204 (talk) 09:13, 12 March 2011 (UTC)

Well, look two pargraphs ago--I think it can be called binary operator: a'a'a=a+3, a'a=a+2 , a'=a+1 ... I didn't understand the use of the sign in the paragraph about complex numbers. 192.116.88.204 (talk) 17:29, 13 March 2011 (UTC)

Illustration of some different methods of interpolation
I've a short treatize in which I visualize a difference between regular tetration, linear approximation and polynomial approximation. http://go.helms-net.de/math/tetdocs/ComparisionOfInterpolations.pdf I didn't see something comparable in the journals, but reading the interpolation-proposals in the article (especially the uncriticized, pompous "linear approximation" as something worth) I think this could give an interesting insight, even if only looked at one of the pictures. If someone of the moderators think it is worth to be linked to (in the article), or one picture should be added, please do so. (If not, I don't mind either) --Gotti 10:21, 10 April 2011 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Infra Logarithm and Ultra Exponential
There's a line in the Super Logarithm section that states that "The infra logarithm function is dual of the ultra exponential function and is denoted by ioga. If a > 1, then it is inverse function of uxpa," and there's a link to the Ultra Exponential function's page... however, that's a redirect to the Tetration page. Is this just someone trying to impose their own names for these functions, or are the "infra logarithm" and "Ultra Exponential" distinct functions? If it's the former, the line should be removed. If it's the latter, then it should be expanded on. — Preceding unsigned comment added by 24.165.127.173 (talk) 00:09, 5 July 2011 (UTC)

Merger proposal
I propose that the inverse function articles super-root and super-logarithm be merged into this article, primarily in the section on inverse functions, but with the names being in the lead. There's less than one paragraph that can be said about each. — Arthur Rubin (talk) 19:38, 29 June 2009 (UTC)


 * Speed Keep These articles should not be merged, because they are simply different functions. I could also merge sum and multiplication because they relate each other. --FixmanPraise me 00:59, 9 July 2009 (UTC)
 * Nonsense. Everything that should be said about those functions is already here.  — Arthur Rubin  (talk) 01:16, 9 July 2009 (UTC)
 * Arthur, please stop saying "Nonsense" to other people, because it may make them offended. Super-root and super-logarithm are expanding, you just have to wait. Also see pentation. Professor M. Fiendish, Esq. 07:27, 25 August 2009 (UTC)
 * The expansion (mostly by addition of material which shouldn't be in the article) should be added, instead, to the inverse function sections of tetration. — Arthur Rubin  (talk) 07:36, 25 August 2009 (UTC)
 * OK, I'm now convinced there may be one paragraph about each, but you're not going to find more. — Arthur Rubin  (talk) 07:42, 25 August 2009 (UTC)


 * super-logarithm seems to have a reasonable amount of content and references published papers indicating that it might be a noteworthy function of it's own (although I haven't read much of the article or checked the sources yet). However, there was very little content in super-root, so I moved it over into the inverse function section and made the page a redirect. User:Robo37 seems to disagree with my merging and undid it (along with undoing a large number of non-contentious edits I made!). Can other people give opinions on this. Is super-root WP:Notable on it's own, is there enough content to make a stub for it? It seems to me that there is not much to say about this function right now and we should have it as a section and only expand it out if enough content and sources are added. Cheers, — sligocki (talk) 08:07, 17 October 2009 (UTC)
 * Also, for what it's worth, User:Professor Fiendish is a confirmed WP:sock puppet. — sligocki (talk) 08:09, 17 October 2009 (UTC)


 * If anything, I'd say that super-logarithms are less notable than super-roots, as more know what roots are than logarithms. And there shouldn't really be any less information about super-roots than there is about nth roots... you could insert a section about operations that can be done with the function or more information about other types of super-roots such as super cube roots and infinite roots, or maybe even something about complex roots... you just need to give the article time to expand. And as Fixman said, they are simply different functions... you wouldn't merge the nth root and exponentiation articles, and in fact square root and cube root both have their own articles even though function wise they are essentially the same thing; and the same applies to the articles about natural logarithms, common logarithms, binary logarithms and indefinite logarithms...


 * I undid all of the other edits you made because they all seemed to be unconstructive and involved removal of large sections of text, if you're planning to remove large sections of an article through a series of consecutive edits it causes a lot less hassle to discuss what your planning to do on the article's talk page first.


 * And Arthur, this picture you're painting of everyone agreeing with you simply isn't true, Sligocki is the only person who agrees with you and that was after some time, after you made the proposal two separate people expressed an immediate dislike to the merge. I'm reverting it for now, but if you manage to get more people to agree with you (I'd say about 2) I won’t mind this merge taking place. And please, don’t turn this into another edit war. Robo37 (talk) 12:24, 18 October 2009 (UTC)


 * May I say, again, "nonsense". Only the super square root has anything said about it.  I have no objection to that having an article, although I doubt notability.  As for "consensus", that is open, but I'm afraid 2 editors, at least one an SPA (that's you), against two established editors, strikes me as leaning toward consensus, and a merge isn't permanent, so "leaning toward" should be adequate.  Sock puppets should be ignored.  — Arthur Rubin  (talk) 13:45, 18 October 2009 (UTC)


 * You've been told to stop saying "nonsense" to people, disregarding someone’s advice and then directing what they told you not to do towards another editor is considered bad practice. And you're saying that there should an article about super square roots, but not one about the super-root function itself? Now if anything, that's nonsense... numbers are likely to have a more varied number of super cube roots - only numbers between 1/e and e have an ∞-order super-root; and in my opinion, complex super-roots are just as notable if not more notable than complex tetration. Something else that's "nonsense" - me being an SPA - I spend most of my time editing articles about games and music, and actually, looking at your contributions, you seem to be more of a SPA than I am. And as for this 'consensus'; if you exclude the guy with two accounts we are still left with 2v2, so the logical thing to do is to not make any changes until the deadlock has ended. Robo37 (talk) 16:05, 18 October 2009 (UTC)


 * When I see nonsense, I comment. We can't (or, at least, we haven't) said anything non-trivial about the super-root function other than the super-square-root.  Perhaps the merger should be completed and super-square-root split out.  (It seems inappropriate to rename super-root to super-square-root, and merge the two sentences over to Tetration; but it would also have been an option.)
 * Perhaps you're not an SPA, although the deleted contribution record may show differently. However, I don't think you can find anything to suggest that I am.  — Arthur Rubin  (talk) 16:19, 18 October 2009 (UTC)

Alright, let's not get into an edit war here.

First of all, Rob-o, if you had looked at my edits you would have seen that they were constructive and I made specific comments for the "large sections of text" that were removed (in several cases this is because they were no longer needed because of the new text). You should not revert edits without looking at them just because you disagree with part of the edit. I took a reasonable amount of time to make the edits I did and I feel that your thoughtless revert was rude. I would appreciate if you took the time to look through it.

Second, as for the actual discussion of super-root and super-logarithm. I think that the main issue here is WP:Notability. The question is: are each of these functions well enough studied to warrant their own article? That is a somewhat subjective question, User:Robo37 and User:Fixman seem to think that they are, while User:Arthur Rubin and I think perhaps they aren't.

Well, certainly many functions deserve their own article, as you note cube root has it's own article even though it is an nth root. However, every function under the sun does not deserve it's own article (for example, there are no articles for fourth root, fifth root, etc.). Why isn't there a fourth root article? Because, in and of itself it is not particularly notable. It is not because there is nothing to be said about these things, for example: I believe that fifth roots can be used to define a formula for fifth-order polynomials, but sixth roots cannot be used to define a formula for sixth-order polynomials. But this could easily be noted in nth roots or elsewhere and it is unlikely that someone will be looking in an encyclopedia for information about the sixth root and find this fact notable.

On the other hand, the cube root is used much more often and has an interesting property that the square root doesn't, there are 3 cube roots of a number and they are not simply negatives of each other, but complex plane rotations. Someone who had to use cube roots and looked that up might find it very helpful to know. Although, the nth root page would tell them the same thing, it's generality might obscure that fact.

So now let's consider super-root, it has been an article since May 2008, yet it is still a stub with no indication of it's notability. There are no mathematical papers cited to suggest that it is actually being studied by mathematicians and there isn't even a note at who coined the term or what one might use it for. It's only notability seems to be that it is an inverse of tetration and since there isn't much content on it, I don't see why it should have it's own article. Is someone likely to be searching for it and be looking for something that they cannot find on tetration?

You claim that not much is known about this function yet, but that as time goes on the article will be added to and expanded to cover what is discovered, but that is sort of a backwards notion of notability. It seems to me that the article should be created when there is something notable to write about, not before. Another concern is that people who see that this is a stub will try to derive a some things themselves just to expand the article, but that is expressly against Wikipedia's WP:No Original Research policy. As an encyclopedia, Wikipedia is trying to preserve accepted knowledge, not create original knowledge. This directly ties to notability. If a subject is not notable enough for anybody to have published an academic paper on it, perhaps it is not notable enough to be included in an encyclopedia. Cheers, — sligocki (talk) 03:57, 19 October 2009 (UTC)

I support the merge. The inverses don't seem notable enough at present, and the useful information about them ought to just go in this article. Joule36e5 (talk) 11:55, 23 October 2009 (UTC)

I do not support the merge. I recall that I found articles concerning the function x^x and its inverse independently of the tetration context. The authors discussed this for formulae in ballistics and some characteristics of driving (google for "wexzal", also I can provide a downloaded copy of that treatize). Also I recall vaguely a study which refer to x^x as function needed for computation of flight of airplanes. However - I cannot correctly recall the actual references. At least I can provide the article on "wexzal", where the author granted me the opportunity to distribute the text freely. So my vote is: get attention for the fact, that "superroot" is a topic independent of the context of tetration; someone with more experience than me may try to find such resources. --Gotti 16:37, 23 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

How about if we do the merge, but keep those two names (super-root and super-logarithm) as links to the appropriate subsection of this article? Then people would still be able to find them by those names, but we'd still have all the info in one convenient place. Joule36e5 (talk) 05:55, 4 February 2010 (UTC)


 * Good idea. That's how merges usually work, but it's best to make it clear.  — Arthur Rubin  (talk) 09:00, 4 February 2010 (UTC)


 * Is the decision made to merge the articles? or is this still being debated? A year is a long time with no conversation, and the last statement was to merge. If no objections soon, I'll start the merge at my sandbox. Cliff (talk) 15:14, 22 March 2011 (UTC)


 * Yep, I think this is a clear merge case. Cheers, — sligocki (talk) 07:24, 23 March 2011 (UTC)


 * No consensus, notable articles. In a website where each little town in Africa is considered notable, a straight forward [used] mathematical function shouldn't need to fight for survival.  Robo37 (talk) 10:36, 23 March 2011 (UTC)
 * It's not used, outside of the specialty (amateur) field. "Tetration" is used in some mainstream journals.  (My other comments that almost that should be in the articles should be here are repeated above, and not disputed, so I'm not going to repeat the details again.)  — Arthur Rubin  (talk) 11:31, 23 March 2011 (UTC)
 * I have to agree that these should be merged. Notability is certainly questionable. Cliff (talk) 18:06, 25 March 2011 (UTC)
 * Robo37, we aren't discussing deletion, simply moving the content to the tetration page. Being concerned with the page's "survival" is a bit extreme since the content will still exist, simply at a different location. Cliff (talk) 22:08, 25 March 2011 (UTC)


 * Regarding the sourses, I've found a lot of stuff on the inverse functions at math.eretrandre.org/tetrationforum/attachment.php?aid=189, and there's cites on the articals as well.. Robo37 (talk) 20:19, 25 March 2011 (UTC)


 * Robo37, I think the sources people in the discussion above were expecting were from mathematical journals. University webpages often have graduate students' dissertations and other original research that has been unverified by the mathematical community at large. I can change the summary to reflect the sources you've found if you think it's necessary (or you can do it). Cliff (talk) 20:29, 25 March 2011 (UTC)
 * Sorry, I see now it's not a university webpage, but still the findings on such a forum are suspect until after a peer-review which usually occurs during and after publication in a professional journal. Cliff (talk) 22:05, 25 March 2011 (UTC)

Merge begun. User:Cliff/sandbox Note, there is additional personal info on the page, please do not disturb. Your input and discussion are welcome. Thanks, Cliff (talk) 22:26, 25 March 2011 (UTC)
 * Merge of super-root complete. Cliff (talk) 19:28, 1 April 2011 (UTC)
 * Super-root page reinstated by user Robo37, citing 'no consensus'. I've reverted the changes and super-rootis now a redirect here. IMO, none of the arguments against merge are valid. the existence/non-existence of other articles does not affect this discussion, and the notability requirement has yet to be met. Further, I believe that original research will be easier to control at the more commonly viewed tetration page. Cliff (talk) 15:52, 4 April 2011 (UTC)


 * merge has been incompletely reverted by Robo37 again, diff here. Robo37, your comment about super root being no less notable than super logarithm is one I agree with, I think both of those articles should be merged to appropriate locations on this page. Your statement does not mean that super root should remain in its own article, but rather that super logarithm is not notable enough to remain in its own article. I understand that that is not the argument you intend to make, but it is, IMO, the only logical way to interpret it. I would appreciate your help in performing the merge of super logarithm to tetration if that is the decision of this discussion.  Cliff (talk) 03:39, 6 April 2011 (UTC)

Now that's out of the way. Let's talk about super-logarithm. Look at the article and decide whether it should be merged here or not. It may have changed since the merge was requested. Cliff (talk) 21:29, 6 April 2011 (UTC)

This merge request is still open. no discussion since April? Does that mean merge? If no objections, I'll start the requested merge. Cliff (talk) 03:41, 13 August 2011 (UTC)


 * No, it does not. 3 for, 3 against is definetly no consensus. Technically the Super-root artical should not have been merged as for large artical chances like this there's first supposed to be a clear consensus gathered, by two votes at the very least, but for now I'm willing to let that one slip. Merging without consensus is a vilation of Wikipedia's principles. See Drop the stick and back slowly away from the horse carcass... Robo37 (talk) 09:29, 13 August 2011 (UTC)


 * Consensus is not determined by votes. We need to determine the arguments for and against merging to determine consensus. Consensus is a term used to describe the will of the entire community, that will is outlined in policy. Consensus does not mean the will of only those involved in the discussion. Cliff (talk) 14:26, 13 August 2011 (UTC)


 * Yes, but there in no real way to calculate that between us, so I've forwarded the discusion to the Dispute resolution noticeboard.


 * Go to Dispute resolution noticeboard for further discusion.
 * The posting at DRN has had no discussion for a while and has been archived as a result. You can find the discussion here. It seems to me that discussion has ended there with most of those responding agreeing that a merge is in order. I'm not sure how DRN postings are supposed to be closed. Any thoughts? Cliff (talk) 19:23, 22 August 2011 (UTC)

Summary
Ignoring Socks.
 * 1) June 2009, Arthur Rubin proposed merging super-root and super-logarithm into Tetration.
 * 2) 3 users have expressed support of the merge (+ one for the merge of just super-root), 3 have expressed opposition.
 * 3) Arguments in support of the merge include; questionable notability, and possible original research
 * 4) Arguments in opposition include; "they are different functions", cube root is an article distinct from nth root, the functions are used (no indication in which fields), and that sources exist (not provided).
 * 5) Merge of super-root to tetration complete. We need more discussion to determine whether to merge super-logarithm. Cliff (talk) 21:29, 6 April 2011 (UTC)
 * 6) Discussion posted at Dispute Resolution Noticeboard by Robo37 on 13 August 2011.
 * 7) DRN discussion archived by a bot due to inactive discussion on 19 August 2011.
 * 8) Posted to RFC by Cliff on 15:27, 20 October 2011 (UTC)

Graph of x^x?
Can someone create a (preferably log-scale) graph of the tetration function
 * $$y = x^x$$

please? It might help me or anyone else visualize how rapidly this function grows!

Thanks,

The Doctahedron, 21:05, 9 December 2011 (UTC)

Super-root use to calculate network clusters
Okay, perhaps I am "dragging a dead cat" as to say as the notability of the function has been brought up many of times before, but I just thought I'd share something interesting on the internet.


 * BOSTON UNIVERSITY COLLEGE OF ENGINEERING - EFFICIENT SELF-ORGANIZATION OF LARGE WIRELESS SENSOR NETWORKS

Scroll down a bit to 'Table 3.1: Worst-case (smallest) cluster size using the Rapid algorithm' on Page 51 one and you should see the following:

And the eW(ln B) collum is given as the super-root of the B.

Robo37 (talk) 17:18, 25 January 2012 (UTC)

rewriting of the iteration-section
Since we intend to show the hierarchy of operations as family of iterations I think it would be more appropriate to rewrite that section with the following example:

$$a$$ and yielding the number after $$a$$, as the 0th):
 * 1) Addition
 * $$a + n = a\!\underbrace{''{}^{\cdots}{}'}_n$$
 * The initial element a is increased by (basis) 1 n times.
 * 1) Multiplication
 * $$a \times n = 0\underbrace{(+a) (+a) \cdots (+a)}_n$$
 * The "additive neutral-element"(0) is increased by (basis) a n times.
 * 1) Raising to power
 * $$a^n = 1\underbrace{(\times a) (\times a) \cdots (\times a)}_n$$
 * The "multiplicative neutral-element"(1) is multiplied by (basis) a n times.
 * 1) iterated Raising to power (no own name for operation was introduced)
 * $$b^{(a^n)} = ((\cdots((b\underbrace{^a) ^a) \cdots )^a)}_n$$
 * The initial element b is raised to the a 'th power n times.
 * 1) Exponentiation
 * $$a^b = $$ (?? - no good idea significantly different from raising to power for a description yet)
 * The initial element b is "exponentiated" with basis 'a'
 * 1) Tetration(iterated exponentiation)
 * $${^{n}a} = \underbrace{a^{\cdot^{\cdot^{a^{a^{1}}}}}}_n $$
 * The "exponential neutral/null-element"(1) is exponentiated with basis a n times.

where I'm not sure whether 1 in Tetration should be taken as neutral or as null element: taken as base-parameter it works more similar to the Null in multiplication and -infinity in increasing/addition; only as top-/start-parameter it is a neutral element (which can be assigned arbitrarily often) in the same way as the 1 in multiplication and the 0 in addition. Also the writing of the powertower/tetration should begin at the top and not at the base.

Gotti 21:05, 2 August 2012 (UTC) Gotti — Preceding unsigned comment added by Druseltal2005 (talk • contribs) 20:47, 2 August 2012 (UTC)

Upkeep
Retracted comment - reason: order of operations failure(talk) 09:08, 6 May 2013 (UTC)

Problems with the extension to infinite heights
The statement that √2^√2^... converges to 2 is just plain false. Seriously, try it out on a caluculator. Heck, just google ((sqrt(2)^sqrt(2))^sqrt(2))^sqrt(2) and you'll find it's greater than 2. — Preceding unsigned comment added by 80.189.71.142 (talk) 18:42, 29 May 2013 (UTC)

But you ignored the right-associativity... try: sqrt(2)^(sqrt(2)^(sqrt(2)^(sqrt(2)^(sqrt(2)^(...))))) on your calculator. — Preceding unsigned comment added by 213.93.217.25 (talk) 13:43, 7 June 2013 (UTC)

Verifying extensions
The section on Extensions contains the following statement - At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. Then the section delves into how such a problem might be tackled. I believe the majority of the section is not verifiable and should be deleted. Daniel Geisler (talk) 20:20, 13 May 2014 (UTC)

¿What is the value of $${}^{-4.3}0.5$$?
First, I show three of the definitions of the tetration for all real "x" and for all complex "a":
 * $${}^{x}a \approx \begin{cases}

\log_a({}^{x+1}a) & x \le -1 \\ 1 + \frac{2\ln(a)}{1 \;+\; \ln(a)}x - \frac{1 \;-\; \ln(a)}{1 \;+\; \ln(a)}x^2 & -1 < x \le 0 \\ x+1 & -1 < x < 0, a = e \end{cases}$$

But ¿What is the value of $${}^{-4.3}0.5$$?

I know it is $$\log_{0.5}(\log_{0.5}(\log_{0.5}(\log_{0.5}({}^{-0.3}0.5))))$$ because of the first formula from the definition I shown. But ¿How would be researched the last argument of these iterated logarithms?

¿Would it be $$\log_{0.5}(\log_{0.5}(\log_{0.5}(\log_{0.5}({-0.3}+1))))$$ (which would be $$\approx 4.03335...$$)?

¿Or $$\log_{0.5}(\log_{0.5}(\log_{0.5}(\log_{0.5}(1 + \frac{2\ln({0.5})}{1 \;+\; \ln({0.5})}({-0.3}) - \frac{1 \;-\; \ln({0.5})}{1 \;+\; \ln({0.5})}({-0.3})^2))))$$ (which would be $$\approx -1.63626 - 3.3882 i$$)?

I believe the value of $${}^{-4.3}0.5$$ should be $$\approx -1.63626 - 3.3882 i$$. Because the base is not $$e$$, is $$0.5$$. Therefore the formula must not be $$\log_{0.5}(\log_{0.5}(\log_{0.5}(\log_{0.5}({-0.3}+1)))).$$ It must be $$\log_{0.5}(\log_{0.5}(\log_{0.5}(\log_{0.5}(1 + \frac{2\ln({0.5})}{1 \;+\; \ln({0.5})}({-0.3}) - \frac{1 \;-\; \ln({0.5})}{1 \;+\; \ln({0.5})}({-0.3})^2))))$$ — Preceding unsigned comment added by DiR3C7 H4CK (talk • contribs) 22:42, 5 June 2014 (UTC)
 * The x+1 only happens if the tetrational base is e, so you would use the quadratic, and the result would be -1.63626 - 3.3882 i.Ashorocetus (talk) 22:19, 11 July 2014 (UTC)

Rational values
A paragraph of the present version of the article reads as follows:


 * It is not known if nq is an integer for any positive integer n and positive non-integer rational q. Particularly, it is not known if the positive root of the equation 4x = 2 is a rational number.

An anon is changing the "exponent" "4" to "2". It has already been known that if x is a (positive) rational number, then $$^2{}x = x^x$$ cannot be rational unless x is an integer. The reference goes on to prove that if $$^3{}x= x^{x^x}$$ is rational, then x is an integer. It's not known for "exponent" 4. Would someone else watch this more closely? — Arthur Rubin (talk) 00:57, 15 January 2015 (UTC)

Is tetration top-down or bottom-up?
I can understand how exponentiation is top-down; $$2^{2^2} = 2^{(2^2)} \ne (2^2)^2 = 2^{(2 \cdot 2)} = 2^{2 \cdot 2}$$ (source: Exponentiation).

But I don't know if $${^{^{2}2}2}$$ is equal to $${^{(^{2}2)}2}$$ or $${^{2}(^{2}2)}$$, because I don't know it's Associative property or Operator associativity.

Can you please expound?

--Indopap (talk) 18:24, 17 January 2015 (UTC)
 * It's "obviously" the former, although this is a matter for pentation rather than tetration. There was, at one point, a discussion of "lower" hyperoperations in that article, but it may have been removed as unsourced.  — Arthur Rubin  (talk) 02:22, 18 January 2015 (UTC)
 * So because $${^{^{2}2}2} = {^{(^{2}2)}2}$$, does that mean $${^{2}(^{2}2)} = {^{(2 \cdot 2)}2} = {^{2 \cdot 2}2}$$ ?
 * I think I found what you were refering to at Hyperoperation.
 * Perhaps "2" is a bad example because $${^{^{2}2}2} = {^{(^{2}2)}2} = {^{2}(^{2}2)} = {^{(2 \cdot 2)}2} = {^{2 \cdot 2}2}.$$
 * --Indopap (talk) 18:11, 18 January 2015 (UTC)
 * I restored Hyperoperation as I found a source using the concept on the ordinal numbers. I eventually unrestored the commentary that "lower hyperoperations" are smaller than "upper hyperoperations" as obviously true, but unsourced.  — Arthur Rubin  (talk) 13:57, 19 January 2015 (UTC)
 * After reading the "Selected values" section of this version of pentation, I can now understand how this works better.
 * --Indopap (talk) 20:17, 19 January 2015 (UTC)

Iterated Exponential Constant
[The https://en.wikipedia.org/wiki/User:Fephisto/sandbox sandbox] page has the beginnings of an article on Iterated Exponential Constants. Fephisto asked me if it should be added to the Tetration page; I don't think so. (Maybe a link could be added.) What do you think?Marvin Ray Burns (talk) 00:09, 18 March 2015 (UTC)

what the..?
what the F is this about? —Preceding unsigned comment added by 201.230.242.190 (talk) 01:07, 29 March 2010 (UTC)


 * Math. Jkar (talk) 02:37, 20 May 2010 (UTC)
 * Does this help? Double sharp (talk) 01:29, 19 March 2015 (UTC)

successor function is more basic than addition
however, the successor function s(a) = a +1 is more basic. Addition is the hyperfunction of succession. this fits the pattern being presented for hyper functions. Counting the cardinal numbers is more basic than addition, as any sesame street graduate knows.
 * the article states that addition is the most basic function.

99.238.16.80 (talk) 13:57, 28 April 2010 (UTC)
 * Know, dear anon, that thanks to your comment I cannot unsee the picture of Count von Tetration. Three ... AH AH AH! Twenty-Seven ... AH AH AH! Seven Trillion Six Hundred Twenty-Five Billion Five Hundred Ninety-Seven Million Four Hundred Eighty-Four Thousand Nine Hundred Eighty-Seven ... AH AH AH! [after a ridiculous amount of time spamming off digits] ... Three Hundred Eighty-Seven ... AH AH AH! [an even more ridiculous amount of time spamming off digits] ... Three Hundred Eighty-Seven ... AH AH AH! [keep repeating that] Double sharp (talk) 07:54, 22 March 2015 (UTC)

Negative exponents
This is wrong

$$ {}^{(-1)}a = \log_{a} \left( {}^{0}a \right) = \log_{a} 1 = 0 $$

The right answer is

$$ {}^{(-1)}a = \log_{a} \left( {}^{0}a \right) = \log_{a} 1 = \frac {i.2.n.\pi}{ln(a)} $$

So, tetration to negative integers is well defined, and trivial. — Preceding unsigned comment added by 186.59.35.113 (talk) 13:59, 27 March 2015 (UTC)
 * You have a point. However, I believe consensus of reliable sources is as we specify.  — Arthur Rubin  (talk) 07:34, 30 July 2015 (UTC)

Mistake in examples table?
I'm guessing this row:


 * $$\exp_{10}^2(3.33931)$$
 * $$\exp_{10}^3(3.33928)$$
 * $$\exp_{10}^4(3.33928)$$

should read

but will let someone who knows for sure make the change. 92.30.45.92 (talk) 11:42, 29 July 2015 (UTC)
 * $$\exp_{10}^2(3.33928)$$
 * $$\exp_{10}^3(3.33928)$$
 * $$\exp_{10}^4(3.33928)$$
 * Actually, I believe the IP is incorrect. Thank you for asking.  — Arthur Rubin  (talk) 07:27, 30 July 2015 (UTC)

In all likelihood the table is correct as written. The values have the form 10^(10^a2), 10^(10^(10^a3)), 10^(10^(10^(10^a4))), etc, where the "a" values are approximated. As we increase the titration height the "a" values approach a limit, and they do so very fast -- but, of course, not instantaneously. That is why the decimal approximations shown look nearly but not exactly equal. Olthe3rd1 (talk) 01:47, 21 September 2015 (UTC)

what is it read as
2+2 would be read as two plus two 2*2 would be read as two times two 2^2 would be read as to to the second power (or possibly two squared) but what would 2 tetration 2 be read as? would it be read as just that, does a term even exist for such a thing? if so what is it? — Preceding unsigned comment added by 76.121.120.23 (talk) 00:08, 5 August 2013 (UTC)
 * I don't think there is anything "official" other than "2 tetrated to 2". Personally, though, I'd say "2 tetrated to the 2nd tower" :) mike4ty4 (talk) 11:14, 10 August 2013 (UTC)
 * I've heard many instances of "A tetrated to the N" and I've also heard "A-tetra-N" and the above "A the the Nth tower", but I've never heard "Nth tetration of A". If I were to vote based on frequency of use, I would pick the first. I'm also a member of the TetrationForum, so I hear tetration in use quite frequently. AJRobbins (talk) 04:00, 7 January 2016 (UTC)
 * I would naturally say "A tetrated to the N" or "A double-arrow N" (following Knuth's notation), though "A to the Nth tower" is so cute that I'm not sure I'll be able to resist it. Double sharp (talk) 06:40, 7 January 2016 (UTC)

Natural Base of Tetration
Why is this article giving examples using e, the natural base of exponentiation? e is a dimensionless number whose ratio to 1 fulfills the following: The rate of change (derivative) of an exponential function of base e is equal to its value. If there is no natural base for tetration, the article should explain why or at least mention it explicitly. R3tex (talk) 00:09, 31 October 2014 (UTC)


 * In fact, there is a "natural base of tetration", it's called $$e_4 \approx 3.08853227$$ (see also Euler number), and the reason why it isn't talked about is because no-one knows it's exact value yet, and it's based on original research, which is forbidden on Wikipedia. Once this becomes more well-known, or known to better accuracy and/or precision, then perhaps it will be allowed on the main page. AJRobbins (talk) 05:23, 12 January 2016 (UTC)

Inconsistent notation.
I am working in my sandbox to merge Super-root into Tetration. As I go I am trying to improve the clarity and coverage of the topic. Unfortunately, I have encountered a problem for which I need input from the rest of the community. The notation for the super-root is inconsistent within each page, and there is further inconsistency between pages. Here are the notations I have seen used and my understanding of their meaning. (If I missed any, please add them) The tetration page does not seem to have a notation for nth super-roots. Instead, each time the function is desired, it is written "nth super-root of x."

Which of these notations is appropriate? Are second and fifth in the table an original notation developed on the Super-root page? Which should be the primary notations on the Tetration page for consistency? (we need one for square super-roots, and one for nth super-roots.  -- Cliff (talk) 21:43, 25 March 2011 (UTC)


 * I think when the section on super-roots was first written, there was no commonly accepted notation, but from what I've seen, $$\sqrt[n]{x}_s$$ appears to be commonly accepted among tetration researchers, now. AJRobbins (talk) 05:35, 12 January 2016 (UTC)

An omission?
From the article:

"Note that exponentiation is not associative, so evaluating the expression in the other order will lead to a different answer:"

"Thus, the exponential towers must be evaluated from top to bottom (or right to left)."

But "A gives a different answer from B" is not actually a sufficient reason for "Thus, the operation is defined as B and not A." Perhaps the reason ought to be explained for the benefit of readers who haven't seen it before. (Evaluating repeated exponentiation from left to right is less interesting, as (a^b)^c = a^bc, so the result is just another exponentiation and not a "new" operation.) 2.24.119.101 (talk) 17:44, 25 January 2016 (UTC)
 * Good point, thanks. I removed 'thus' for starters, which was logically wrong as you pointed out. I'm not sure if more explanation is necessary, but feel free to do so. Gap9551 (talk) 00:41, 26 January 2016 (UTC)

Problem with this article being quoted as authoritative on tetration
Google 'site: math.stackexchange.com "wikipedia.org/wiki/Tetration"' to see a number of references from math.stackexchange.com for the Wikipedia's tetration article. The problem is that material in the real and complex extensions to tetration liberally publish material I've never seen published before and it has no citations. Daniel Geisler (talk) 01:18, 8 February 2016 (UTC)

Euler's theorem for last m digits
I added a clarify tag to the passage "When a and 10 are coprime, we can compute the last m[clarification needed] decimal digits of $$^{n}a$$ using Euler's theorem."

What does m represent in this context? I do not know enough about the topic to clarify it myself. I suspect that m might be meant to be n? Evenhash (talk) 21:28, 16 January 2017 (UTC)
 * It's an arbitrary integer. For example, consider Graham's number, which is a very very large 3-tower.  We don't have any way to examine its first digit, but we can easily compute, say, the last 12 digits, as is done on the page for that number.  Anyway, I've made the clarification in the text, and removed the tag.  Joule36e5 (talk) 08:32, 26 February 2017 (UTC)
 * Well, it is quite possible to examine the first digit if you get to choose the base! ^_^ In binary it must start with 1; as it is a power of 3, it must also start with 1 in ternary, and look like a 1 followed by a lot of zeroes (and I'm aware that "a lot" is quite the understatement). In fact, since it is an odd power of 3, being 3 to the power of some almost-as-very-very-large 3-tower, in base 9 (novenary) its first digit must be 3. And so on. Double sharp (talk) 10:24, 26 February 2017 (UTC)

0↑↑∞
= 1/2 ? --2.247.251.5 (talk) 06:16, 2 November 2017 (UTC)
 * Why? Even if you choose to set 00 = 1, then n0 is 0 for odd n and 1 for even n anyway, so the sequence (n0) doesn't approach anything. It could be considered another indeterminate form, I guess. Double sharp (talk) 12:32, 2 November 2017 (UTC)

order of indexes in nested exponentials and infinite iterated exponentials
I'd like to repeat my criticism against the usual notation of indexes for the mentioned operations.

We find in the article:


 * {|class="wikitable"

! Form

! Terminology
 * $$a^{a^{\cdot^{\cdot^{a^a}}}}$$
 * Tetration
 * $$a^{a^{\cdot^{\cdot^{a^x}}}}$$
 * Iterated exponentials
 * $$a_1^{a_2^{\cdot^{\cdot^{a_n}}}}$$
 * Nested exponentials (also towers)
 * $$a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}$$
 * Infinite exponentials (also towers)
 * }
 * $$a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}$$
 * Infinite exponentials (also towers)
 * }
 * }

where it should be


 * {|class="wikitable"

! Form

! Terminology
 * $$a^{\cdot^{\cdot^{a^a}}}$$
 * Tetration
 * $$a^{\cdot^{\cdot^{a^{a^x}}}}$$
 * Iterated exponentials
 * $$a_n^{\cdot^{\cdot^{a_2^{a_1}}}}$$
 * Nested exponentials (also towers)
 * $${\cdot^{\cdot^{a^{a^x}}}}$$
 * infinitely iterated exponentials
 * $$\cdot^{\cdot^{\cdot^{a_3^{a_2^{a_1}}}}}$$
 * Infinite exponentials (also towers)
 * }
 * infinitely iterated exponentials
 * $$\cdot^{\cdot^{\cdot^{a_3^{a_2^{a_1}}}}}$$
 * Infinite exponentials (also towers)
 * }
 * }

The adaption of the order of indexes seems so natural to me, that I'm afraid I'm overlooking some special reason for that current use. For instance, the order of indexes which I claim to be wrong, can be found in Gennadi Bachman, "Convergence of infinite exponentials" (and in many more places. In the article of Ernst Schröder "Über iterirte Funktionen" (1871, Math Annalen) he begins his text with a recursive definition which suggests the non-inverted indexing scheme). However I cannot imagine one reason for this common (apparent inverse) indexing-scheme. So let me state my concern again.

We begin the operation at the top-level, so that should have index 1. ALternatively it should contain an initial value x from where the iterated operation starts (in the case of iterated exponentials, for tetration we assume x=1).

While this notational difference may be seen as neglectable for the finite case, this is not neglectable in the infinite case of iterated exponentiation. If the top-level is the dotted continuation, then there is no initial value x different from the iterated base a from where the operation could have started. If the top-level has x, and going to the bottom I assign an exponential base a to the previous value then I can eventually formally write the dots to the bottom, indicating, that I mean the infinite continuation of appending exponential bases.

Second: from this I'm tending to not to talk of "powertowers" here, but of "exponential-towers", since "power-towers" sounds more that I start with a value x, raise it to a power a, raise that to a power a and so on. While in tetration/iterated exponentiation I start with a value x, give it an exponential base a, give that an exponential base a and so on.

Schröder:Über iterirte Funktionen

Bachman:Convergence...

Gotti 18:21, 8 October 2009 (UTC)


 * I was also under the same impression until I had an epiphany. Consider that these towers are to be used in a fashion similar to sums and products, in which case we would want them to have interesting convergence properties in their own right, for lack of a better word, "interest". Let us consider the sequences associated with these towers, and see how they relate.
 * $$\cdots \uparrow a_3 \uparrow a_2 \uparrow a_1$$
 * would have the associated sequence
 * $$\{a_1, a_2^{a_1}, a_3^{a_2^{a_1}}, \cdots\}$$
 * and no matter how interesting this sequence is, the limit should always be $$(a_{\infty})\uparrow\uparrow\infty$$. Now this is just a conjecture, but I would imagine that a proof is possible. The point is that this view of towers has an associated sequence whose limit makes no use of towers, and is simply expressible with infinite tetration (or the H function). Just as "iterated powers" are usually dismissed as too trivial to be of interest, and "iterated exponentials" as more interesting because they cannot be reduced, I think the same reasoning applies to this view of towers. The one redeeming quality of this sequence is that each member of the sequence can be calculated by performing a single exponentiation on the previous member, which certainly saves time.
 * Now consider the other view of towers:
 * $$a_1 \uparrow a_2 \uparrow a_3 \uparrow \cdots$$
 * which would have the associated sequence
 * $$\{a_1, a_1^{a_2}, a_1^{a_2^{a_3}}, \cdots\}$$
 * This view of towers is much more rich and complex, and has much more room for unexpected convergence conditions. In short, it has the "interest" that we were looking for. Granted, it is much harder to calculate, because every member of the associated sequence must be calculated from scratch, so we cannot use the previous member to help in calculation, but that is the price we pay for "interest". AJRobbins (talk) 07:52, 15 October 2009 (UTC)
 * Yes, this approach looks to yield more behaviors. Consider doing this with $$a_n = 2^n$$ and $$a_0 = 2$$. Then $$\{a_0, a_0^{a_1}, a_0^{a_1^{a_2}}, \cdots\}$$ approaches a 2-cycle. mike4ty4 (talk) 21:35, 10 December 2009 (UTC)
 * Mmm. Looks like one can get this type of behavior with the other order of tower as well (even with mixed exponents). mike4ty4 (talk) 22:34, 10 December 2009 (UTC)

Hi Andrew! ("gotti" - that's me Gottfried) —Preceding unsigned comment added by Druseltal2005 (talk • contribs) 11:56, 19 October 2009 (UTC) Hmm, I may understand correctly what you mean with "more rich and complex", at least I have an idea, and as a "lover of structures" this could also be an argument for me. However there is the non-vanishing argument for the indexing beginning at the top for the infinite case: you don't have an initial value for the iteration other than the base itself. And for the fact that 2^(1/2) = 4^(1/4) we had/have no notation using the exponential-tower, if/because the top index is shifted to infinity. But actually, I cannot imagine any branch of number theory, where we would not begin the index-count at that position where we begin to compute. So all this discussion makes me the more headscratching...where I might have overlooked some important thing (or am unable to understand)... Gottfried --Gotti 11:55, 19 October 2009 (UTC)

I don’t think we should shy away from the word ”powertower”. 3^3 it pronounced ”three to the POWER of three” for shure. And that can be further extended and then we get tetration or powertower. To ask oneself wether it is ”iterated powers” or ”iterated exponentials” is not relevant since the article is not about a function but a (binary) OPERATOR that takes two operands and does some number crunching. Certainly y=x^3 is a power FUNCTION, and y=3^x is an exponential FUNCTION. But again, what we are discussing is an operator. Properly typset 3^^5, to me at least, looks like ”a tower of powers”, a powertower. And from there you can’t go wrong since without any brackets it is supposed to be evaluated from the top-right to the bottom-left.83.223.9.100 (talk) 13:39, 9 March 2018 (UTC)

I looked into the article of L.Euler "De Formulis Exponentiabilus Replicatis" (E489 in Eneström index). Euler makes it explicite, that he discusses the exponential-tower in the following way: a, b=r^a, c=r^b (he uses greek letters) and evaluates always from the top. Gottfried --Gotti 06:18, 7 October 2010 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Pronumciation
...and can somebody add a pronunciation guide for the different notations?

Eg: a plus b, a times b, a to the power of b, a something something something dark side b. — Preceding unsigned comment added by 82.38.206.148 (talk) 05:19, 19 May 2012 (UTC)

Whats with the Format

 * The opening paragraph seems a little imbalanced. Their are double spaces where they don't need to be. Am I overlooking something? — Preceding unsigned comment added by Mulstev (talk • contribs) 19:15, 21 December 2018 (UTC)
 * Do you mean that the lines look uneven? That is probably because of the mathtext; things like $$a^{a^{a^a}}$$ require higher than normal text roofs, so the entire line drops down to accommodate it. I'll try to think if there is a way to fix the line spacing while retain the information in the lead. Integral Python click here to argue with me 20:46, 21 December 2018 (UTC)

Sums of the reciprocals of a↑↑b
For given b:

For given a (starting from n = 0):

ArtismScrub (talk) 01:39, 14 March 2019 (UTC)

Holomorphic tetration
In which namely domain is it holomorphic? The graph appears to show a snip somewhere like $(−∞, −1]$. Incnis Mrsi (talk) 09:04, 2 May 2019 (UTC)

Moving towards a verifiable article
It's been five years now since I raised the issue that the material on fractional iteration appears to only exist in only two places, on my web site Tetration.org and on on Wikipedia's Iterated Function page. I requested a reference from the author because I know my mathematical work when I see it. The author claimed a reference was not needed. Note the author did not claim to the originator of the material. This is listed as an article of medium mathematical importance, but no one else has ever had and issue with the amount of unverified and unattributed material. So I'm being proactive and fixing things. I am also starting by only adding material and not deleting anything.

The material on fractional iteration on both the Iterated functions page and the Tetration page are both only not verified, they are not attributed. This is in direct opposition to the Wikipedia policy on only having verified content. I've raised this issue for many years on the Iterated functions and the Tetration pages and it has only fallen on deaf ears. I am a primary researcher on the subject and my web site Tetration.org has been up in close to its current form for fifteen years. The site has been "peer-reviewed" by Micheal Somos, Ioannis Galidakis, Andrew Robbins, Ed Pegg Jr, Stephen Wolfram, Ruben Aldrovandi and Rudy Rucker. Only a few mathematical flaws have been found in over fifteen years.

Daniel Geisler (talk) 02:53, 30 April 2019 (UTC)

Qustion About Being a Credible Source

 * So, just to be clear before I post a full response to this, you are essentially saying that there is a lot of unsourced material in this article, and that this material comes from tetration.org, and that tetration.org should be considered a credible source? Correct me if I'm wrong or missed anything.


 * Integral Python click here to argue with me 10:20, 30 April 2019 (UTC)

Reply
Precision is important here. Point by point. The problem of "a lot of unsourced material in this article" is absolutely true. I was an early editor of the tetration page and had written much of its content in 2007. Then the article received heavy editing by people with a little apparent mathematical background in the subject. Many of them felt they had something important to add by sharing their personal notation for tetration. The article became a dumping ground for bad mathematics and I exited my editorial involvement in Wikipedia. What gets published on the tetration page is solely determined by politics, not Wikipedia policy. I raised some issues five years ago and in that entire time, there has not been a single response. It was very frustrating to see a mathematician of John Baez's stature referring to a portion of tetration page that was unattributed, unverified and just plain wrong! Mathematically tetration is a subset of complex dynamics. Proofs regarding tetration that are inconstant with established research in complex dynamics are invalid.

Material coming from tetration.org - I monitor developments in extending tetration, so I try and be familiar with all published research. I have only seen the material in Iterated_function for the Iterated Functions page, which is directly relevant to extending tetration, in one place, my web site. I asked the author for a reference and I was ignored. I feel either the author needs to provide a peer-reviewed reference for the material or the material such be deleted to be consistent with Wikipedia policy. I'm not very impressed with the argument that the current material is "cool" and removing unattributed material would shrink the article too much.

Tetration.org should be considered a credible source - please note that the title of my posting is "Moving towards a verifiable article". My website has been cited in several published papers. The content has been pretty stable for fifteen years and I've only been notified of three minor math errors in that time despite recruiting as many major mathematicians to review the site as possible. There is an unimaginable amount of inertia in publishing material in fractional iteration. My understanding of the virtual impossibility of publishing in this area and my own research is the following. Henri Poincaré was unable to place the unification of maps and flows, of discrete and continuous dynamical systems on an algebraic foundation and so he developed topology. My work is about repairing that rift. My method of unifying maps and flows is based on elementary mathematics. So if my research is valid, I could have published my work one hundred and fifty years ago and had it understood.

Daniel Geisler (talk) 18:45, 30 April 2019 (UTC)


 * There are issues of Notability and Reliable source here. If the fractional iteration material exists in only one place outside Wikipedia, then that place is the primary source, so there are no secondary sources, which are Wikipedia's bread and butter.


 * To put it in a different way: Wikipedia's job is not to move mathematics forward. Wikipedia's job is to document established, notable human knowledge. Mathematics that is too close to the cutting edge is specifically not for Wikipedia, even if it is eventually known to be correct and important. So I worry about whether these contributions are currently appropriate.


 * Please let me know if I've misunderstood. And please notice that I am not commenting on whether your math is valid or important. Mgnbar (talk) 20:26, 30 April 2019 (UTC)


 * I mostly agree with Mgnbar. While I am sure that there may be secondary sources for many of the things mentioned by Daniel, I think there is probably also a fair amount of primary work. Daniel, your work at the Tetration forum sounds interesting, but if your ultimate goal is to just have the article be what you have discovered on that forum and credited to it, there wouldn't be much of a difference between the two, would there? As for your lengthy speech about the injustices against mathematicians with disabilities, I am not an expert on that and I don't think it has much relevance to the topic anyway. I am afraid you have as yet to convince me that Tetration.org is a good source to use on Wikipedia (note that I said "good", not "reliable"; I have used the website for referencing on the tetration many times and trust it very much so).


 * I will look into parts of this article which appear to be unsourced or sourced to a single primary source and see what I can do about them, and I invite you and any others to help me. Integral Python click here to argue with me 00:17, 1 May 2019 (UTC)


 * Daniel, I would also like to point out that you are referenced in the references section, just not in inline citation formatting. Thanks, Integral Python click here to argue with me 00:19, 1 May 2019 (UTC)

Thanks for the feedback people. Stephen Wolfram weighted in with the following remark "... though I (sadly) haven’t had time to study your work in as much detail as I would like, it seems very interesting and elegant, and deserves to be better known."

Here is the fundamental issue. If people supporting a strict interpretation of no unpublished material on Wikipedia then I will be happy to go through the Tetration and Iterated function pages and purge them of all unsupported material. Also please don't try and peer into my soul and to see what hidden agenda I have, unless you can provide a Reliable source for knowledge of what I'm thinking. It's also a bit paradoxical to insist on substantiated information on my part while sharing your attempt at reading my mind. Lets keep this on an objective level.

Please weigh in on whether you support my removal of all material that is not Notability or a Reliable source. I will no longer support having different criteria for my material and the current material, as has been the case for the last twelve years. Let's replace politics with policy.

Daniel Geisler (talk) 07:49, 1 May 2019 (UTC)


 * First off, please stop talking about "replacing politics with policy." I believe you have no hidden agenda and never thought you did. No one did, and it seems you either made it up or assumed I had some hidden agenda to discredit you, which would be hilarious ngl but that's not the point. As far as I have seen on this article, politics has not been much of an issue here and your claim otherwise seems to be based on nothing we have talked about. Furthermore, no one has said that your claim isn't reliable and no one has said you are wrong. Yes, this article has a problem with original research and needs better citations, particularly in the real and complex extensions section. No, there is no evil organization set on discrediting you and filling the article with uncited work. As far as I can see, recent edits to the page involve a few vandals (and reverters), an IP, and me, with a few others making occasional edits. None of the edits attempt to add any unreferenced material or remove reliable sources. This whole argument is a non-issue, as no one has ever not supported the wikipedia policies.


 * Lastly, it doesn't matter whether anyone "supports" the Wikipedia guidelines in this instance. Wikipedia is not a democracy. It seems to me that you could have just went in and started tagging things with Citation needed and no one would have even looked at it twice, but instead you went to the talk page and didn't even make a single edit to the article itself.  Be bold, dude.


 * Finally, to address your original original point about fractionally iterated functions and original research specifically, that has nothing to do with this artical even besides the fact that I have seen plenty of papers about it not in either of those two websites. Also, anything from Stack Exchange is not in itself very notable so has nothing to do with the conversation.


 * Sorry if I'm rambling a little. In summary: Yes, OR and Notability are problems. Stop talking about nonexistent politics. No, a vote isn't needed. No, if tetration.org is the only source then it should not be on wikipedia, not correctly attributed. Lastly, No, there is no cabal. Oh yah, and while you could probably write a book about it and it seems very interesting, putting your personal life on wikipedia had nothing to do with the conversation at hand. Integral Python click here to argue with me 12:01, 1 May 2019 (UTC)


 * Daniel Geisler, I agree that Wikipedia is inconsistently sourced, with a lot of old material appearing to be protected under some grandfathering policy that doesn't actually exist. ;)
 * Rather than delete swathes of material, please first place Citation needed tags on the material that you would delete. That would help your fellow editors understand the problem better.
 * You might also want to read Scientific citation guidelines. They are liberal when it comes to formulating examples, for instance, although that might not be the problem here. Mgnbar (talk) 12:47, 1 May 2019 (UTC)

I do appreciate the links provided to pertinent Wikipedia policy, as I have been out of the loop for a decade. I will study them and modify my content to be consistent with them. I have stated that my intent here is to unify people, not to start a war. So I would like to begin by finding the area of agreement that we all have. I have removed the personal content from this page that people found inappropriate because their concerns were totally valid.

I'm sorry, but I am very dissatisfied with the editing on these two articles. I don't think that the editorial issues are resolvable with the current group of folks checking in now. So I have promoted it up to the Wikipedia_talk:WikiProject_Mathematics page. I want to hear from folks with arguably no prior bias and greater editorial responsibility that the current editors of these two articles. I think it is completely appropriate to elevate this conversation to beyond this page.

Please answer the following questions. Why in five years has there been no response to my editorial concerns? Why do folks insist that I buy into their idea that there are no politics going on here? It is clear that people are not enforcing Wikipedia policy, but no one wants to own up to it. Not to bemean, but why am I'm required to believe that I am talking to a group of saints? Once again, I am only looking for people to back a consistent treatment of my edits with the edits of others.

Would it be appropriate and useful for me to edit a copy of the two pages on my user page?

Daniel Geisler (talk) 21:00, 1 May 2019 (UTC)


 * Hi again, Daniel Geisler. I arrived here after seeing your appeal at Wikipedia talk:WikiProject Mathematics. I have no prior experience with these articles or you (as an editor or mathematician).


 * I do not know the answers to your questions, but I'll try: Perhaps there has been no response to your concerns because there aren't enough active editors paying attention, or your fellow editors expect you to Be bold and fix the problems yourself. I don't see any evidence of excessive politicking or abuse, so I don't understand your question about politics. You are not talking to a group of saints --- just fellow volunteers, who are imperfect people with limited time and attention for any one issue.


 * You are free to edit copies of the two pages in your user space, but I don't think it will lead to any satisfaction for you, because there will be friction if/when you try to re-integrate them back into the main space. Usual Wikipedia practice is to make many small-to-medium edits on the main space page. Why? Because then other editors can track your progress easily in the page's history.


 * You have not responded to my concrete suggestion that was intended to help you: Place citation-needed tags on the contentious content, so that newly arrived editors such as myself can better understand the contention. What do you think about that? To me it is the obvious first step toward your remedy.


 * Although I understand why you did it, please do not edit your posts on a talk page after other editors have responded to them. Why? Because doing so alters the context and therefore the meaning of the other editors' posts. Mgnbar (talk) 21:22, 1 May 2019 (UTC)

Thanks for the input, your comment on using citation-needed tags sound like exactly what is needed. I have chosen not to be bold because back in 2007 I added a bunch of content and a large part of it was soon edited out of existence by content that I could not support. Many people feel that the proper way forward is through publication on peer-reviewed journals after which the content on Wikipedia can be added as is appropriate. That is my focus. My agenda here is simply to be a good academic citizen. I am more than willing to work with the people interested in editing these pages by providing support when requested. Also, my math skills eclipse my people skills, so I'm not the right person to drive the overall editing process. I would love to hear what direction others would like to take these pages. If nothing else, the graphics and fractal images I've shared have been popular. I'm working on tools to help people visualize the dynamics of the exponential map including making a movie with Mathematica. Thanks.

Daniel Geisler (talk) 03:49, 2 May 2019 (UTC)

I am sorry, but this is not the appropriate place for me to spend my effort.

Daniel Geisler (talk) 08:04, 2 May 2019 (UTC)


 * I'm sorry that you have given up, but your frustration is understandable. To sum up my involvement with the article: I still do not know which content is contentious, so I can neither source it nor delete it. Best wishes. Mgnbar (talk) 11:03, 2 May 2019 (UTC)

Thank you for your support Mgnbar. I'm not really giving up. My job as a researcher is to publish in peer-reviewed journals and to become a more effective educator. So the best way I can do good for Wikipedia is outside of Wikipedia.

Daniel Geisler (talk) 15:07, 2 May 2019 (UTC)

More footnotes
This article has non-inline citations and is currently B-class. To attain GA status, more references should be inline, particularly in the "Real heights" subsection. Thus, I have added a more footnotes tag, even though no potentially objectionable statements are unsourced. I have also moved one citation duplicated by a footnote to a new "Further reading" section, as is standard for this source. After this, it may meet the other GA criteria as well. – Laundry Pizza 03  ( d c&#x0304; ) 23:22, 5 July 2019 (UTC)

Fifteen year old errors
The tetration article and others are considered untouchable. I've been unable to make any significant changes in ten years. I used the Wayback Machine and found fifteen year old errors. See User:Daniel_Geisler/Tetration for a lean version of the tetration page.

Daniel Geisler (talk) 19:05, 26 December 2019 (UTC)
 * May I ask to you why you consider this article to be untouchable? I have done a fair amount of editing here in the past and encountered only minor resistance for some of my edits. As for your version of the page, it seems rather similar except missing the majority of the extensions. You may have done so before, but why do you believe those fairly heavily researched sections shouldn't be on the page? Integral Python click here to argue with me 17:51, 31 December 2019 (UTC)
 * I had too many edits reversed. Yes, I also had "only minor resistance for some of my edits", just not the consequential edits. "Heavily researched"? I have fifty years research in tetration, isn't that heavy? But the Wikipedia criteria is for published research.

Daniel Geisler (talk) 23:47, 4 January 2020 (UTC)

A whole method for tetration has been deleted ! Why ?!
Recently I added a pretty detailed method for tetration. It is quite famous in the tetration community such as the tetration forum and well accepted. Why has it been removed ?!? Tom.Raes (talk) 22:44, 21 January 2020 (UTC)
 * "and well accepted"
 * In more detail, a forum is considered a self-published source and therefore is unreliable. We need a reliable source, preferably a peer-reviewed journal article.--Jasper Deng (talk) 23:23, 21 January 2020 (UTC)
 * There are no journals dedicated to tetration and the tetration forum is ofcourse the best source for tetration. It is the ONLY tetration community online and it is moderated by a math prof who published about tetration. In the link it is confirmed by prof H Trappman that the method works. Tom.Raes (talk) 23:41, 21 January 2020 (UTC)
 * There are no journals dedicated to tetration and the tetration forum is ofcourse the best source for tetration. It is the ONLY tetration community online and it is moderated by a math prof who published about tetration. In the link it is confirmed by prof H Trappman that the method works. Tom.Raes (talk) 23:41, 21 January 2020 (UTC) Tom.Raes (talk) 23:46, 21 January 2020 (UTC)
 * Doesn't make it any more reliable (professors can also be wrong). There are journal articles cited in this article, so I don't see why you couldn't provide one here.--Jasper Deng (talk) 00:16, 22 January 2020 (UTC)

((a^a)^a)^a
Is there a term for the opposite (left to right) order of precedence, i.e. repeatedly raising to a constant power? Cesiumfrog (talk) 02:54, 25 May 2020 (UTC)


 * Probably not because it would simplify to a^(a*a*a) = a^(a^3). Cesiumfrog (talk) 05:28, 25 May 2020 (UTC)

The superimaginary?
Define a surreal constant s as ssrt(-1). Is it possible to break this down into real/imaginary parts, or would it turn into its own unique unit? ArtismScrub (talk) 21:34, 13 March 2018 (UTC)
 * $$s=-1$$, as $$^2(-1)=(-1)^{-1}=-1.$$. So it's in fact real(!). Alfa-ketosav (talk) 12:18, 1 July 2020 (UTC)

Open questions
The article has an outdated sentence: "In particular, it is not known whether either of $4&pi;$ or $5e$ is an integer." In 2020, modern computer algebra systems are able to answer one of these questions, namely that $4&pi;$ is not an integer: $$\begin{align} \pi^{\pi^{\pi^{\pi}}}\approx 9.08\times 10^{666262452970848503} \end{align}$$ 213.41.135.21 (talk) 08:44, 12 July 2020 (UTC)
 * That doesn't prove it's not an integer; it only proves that if it is an integer it has 666,262,452,970,848,504 digits (that makes it big!) Georgia guy (talk) 11:40, 12 July 2020 (UTC)
 * But has it not been recently proved (by example of the first Skewes`s number) that the result of multiplying two transcendental numbers cannot be an integer? 213.41.135.21 (talk) 15:23, 12 July 2020 (UTC)
 * If that statement were true, it would prove that there are no transcendental numbers. Two transcendental numbers CAN make an integer when multiplied together; an example is pi times (1/pi) equals 1. Georgia guy (talk) 15:29, 12 July 2020 (UTC)
 * I put it a little inaccurately. I meant the exponentiation (1/pi)^(1/pi) is not an integer 213.41.135.21 (talk) 15:41, 12 July 2020 (UTC)
 * It clearly isn't; it's approximately 0.69462799224. Georgia guy (talk) 15:50, 12 July 2020 (UTC)

Euler paper on infinite height
How come that this paper is not mentioned: De formulis exponentialibus replicatis https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1488&context=euler-works — Preceding unsigned comment added by Laczkó Dávid (talk • contribs) 19:07, 22 August 2020 (UTC)

Examples correct?
I think there is an error in the table of "Examples of tetration", I find it suspect that for x=3,...,9, when x^^4 = exp_10^3(z), then x^^5 = exp_10^4(z) (??) which is 10^exp_10^3(z) = 10^(x^^4), while x^^5 = x^(x^^4), and 10 &ne; x. &mdash; MFH:Talk 13:56, 17 February 2022 (UTC)
 * Wait... x^^5 = x^(x^^4) = 10^y with y = x^^4 lg x = exp_10^3(z) lg x = 10^t with t = exp_10^2(z) + lg lg x, and OK, lg lg x ~ -0.3 is small w.r.t. 10^10^z but not completely negligible for z = 1.09... but yes, it won't change the z-value within the given precision. This might merit a remark: For numbers x &in; [2 .. 99] and n > 2, we have x^^n = exp_10^m(z) => x^^(n+1) = exp_10^(m+1)(z') with |z' - z| < 10^(-???). &mdash; MFH:Talk 15:18, 17 February 2022 (UTC)