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I've read a passage in the forum about tetration and did some research on Wiki. I understand the basic definition for any real height $n>-2$,

$$^na=a^{a^{a^{a}}}...\text{, for real height =n}$$

I also know that value tetrations with fractional height $n>-2$ can be computed and $^na$ is a continuous function.

However, my question is:

What is the physical meaning of Tetration with fractional height?

e.g. $^22=2^{2}\text{ , }^32=2^{2^{2}}$, but how come there is a $^{2.5}2$ ?

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    $\begingroup$ people have worked on this for a long time, there is no evidence that such an operation is possible. The goal would be something along the lines of the Gamma function, which interpolates the factorial and is meromorphic. $\endgroup$ – Will Jagy May 9 '15 at 17:58
  • $\begingroup$ By the way, do you mean $n \geq 1$ versus $n > -2$? $\endgroup$ – Simon S May 9 '15 at 18:24
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    $\begingroup$ @QiaochuYuan I remember an old answer of you at MSE explaining this point of view that is somehow linked with the discussion "It Ain't No Repeated Addition" (.. by Devlin right?). Anyways I was thinking if this implies that... the "iteration relationship" between the hyperoperations (Goodstein's ones) is only a curious coincidence... I'm not expert at all but when you say "Tetration has no physical meaning." I'm tempted to add "yet". The hyperoperation sequence definition seems too weird/nice to be a coincidence, MAYBE we have not yet found a role for higher HOs in the universe. $\endgroup$ – MphLee May 10 '15 at 14:15
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    $\begingroup$ @MphLee: I find your comment well written, and the keyword "universe" reminds me of some (so far fruitless) speculation, whether the problem of the hypothesized extreme dilatation in the near of the big-bang might be a candidate for mathematical models involving iterated exponentiation . A second, even more, speculative idea was when the evolution of the time itself was discussed. Again this has for me some flavor of selfcomposed functionality. There is also an electronic tool, the "avalange diode". Perhaps its curve is already perfectly modeled- otherwise that would be a candidate to look at. $\endgroup$ – Gottfried Helms May 11 '15 at 7:14
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    $\begingroup$ @QiaochuYuan, found the reference. I'm talking about this math.stackexchange.com/questions/35598/… $\endgroup$ – MphLee May 11 '15 at 13:02
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Lars Kindermann in his dissertation discusses fractional iteration with the following focus:
Consider the production of sheets of steel in a sequence of similar machines (pressing and bowing that sheets to bring it into form), say, seven such machines behind each other. Then you can measure the sheets before the first and after the last, but you cannot (by some technical reason) measure their progress. Kindermann says then that the goal is to interpolate the forming process mathematically, and that the formulae required that of iteration, and then that of fractional iteration (iteration-height = 1/7); however he approached the mathematical modeling with a neural network not with an analytical formula (his dissertation is online but unfortunately only in german language, but has also lots of references). Kindermann's list of links

A second example is mentioned by the russian physicist Dmitri Kouznetsov where he says that modeling the flow in an optical fibre depending on the fiber-length needs fractional iteration, and he provides an approach for fractional iteration of the exponential and other functions (His main article is linked in the wikipedia-article on tetration, and has also been printed in some journal, and he has a compressed info at the "citzendium"-wiki)

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