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Tetration is a natural extension of the concept of addition, multiplication, and exponentiation.

It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest hyper-operations, as they are called.

For example:

  • Adding the forces on an object to find the resultant force.
  • Multiplying the length, width, and height of a box to find the volume.
  • Using exponentials to describe the decay of radioactive atoms.

But are there any phenomena in the natural world which are simply modeled by an equation involving tetration, some higher order hyper operation, or their inverses (such as the super-root or super-logarithm for tetration)?

And if not then why do natural phenomena only seem to behave in ways which can be modeled by hyper operations of order 3 or lower?

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2 Answers 2

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Tetration is a natural extension only of the integer-valued notions of addition, multiplication, etc.

For example, the notion that multiplication is repeated addition fails when you start multiplying by non-integers, and even if you hack together an explanation for the rationals, it requires even more care for the irrationals. The same is also true for exponentiation.

Most things that use these three operations in the physical world involve real-valued operations. For example, your power-law for a fluid viscosity might be $v^{2.4}$, or pressure is related to the inverse of the area, and so forth. Obviously, $e^t$ and its variants appears all the time.

Tetration, however, does not have a unified definition for real or complex heights. As such, its application to our physical domain is quite limited only to areas where things are always integers. And in physics, that happens quite rarely.

Edit: That's not to say that there couldn't in principle be something that is a real number tetrated to an integer height. But -- and there's always a but -- we have to account for units. If you take a length measurement and raise it to the 3rd power, it becomes length-cubed. If you take a length measurement and tetrate it to the 3rd, then it becomes...???

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    $\begingroup$ One should always mention, that the connotation of an operation of higher order as a repeated application of the one of lower order might be only an initiating/motivating model. After having established that iterated application (and having a set of properties/rules) the interpolations to non-integer iteration-"heights" is natural and brings up connotations, which do no more simply model iteration, but have a new quality. This discussion is not new see en.wikipedia.org/wiki/Multiplication_and_repeated_addition and especially the very readable article of Keith Devlin in AMS. $\endgroup$
    – Gottfried Helms
    Commented Jul 25, 2013 at 9:55
  • $\begingroup$ @Gottfried Helms Thanks, I read the wiki article and I found it very interesting, the debate highlights very well the non-triviality of the Hyperoperations research!! Anyways I can't find the Devlin's article because it is not available anymore, Have you a link. $\endgroup$
    – MphLee
    Commented Jul 25, 2013 at 12:29
  • $\begingroup$ @MphLee: it is just about one week that the article seems no more available - don't know the reason for the retraction. I'll see whether I can do something instead... $\endgroup$
    – Gottfried Helms
    Commented Jul 25, 2013 at 15:30
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    $\begingroup$ @MphLee: try this at the web-archive: web.archive.org/web/20120922150415/http://www.maa.org/devlin/… $\endgroup$
    – Gottfried Helms
    Commented Jul 25, 2013 at 15:51
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This seems to be a duplicate of What is the geometric, physical or other meaning of the tetration? From the earlier answer of mine:

There is an entry in citizendium, where D. Kousnetzov describes his proposed general solution for the tetration-function. He links to some papers of his own where he gives more examples (there are only few so far) of physical applications. (Something with light transmission in glass-fibers, increasing mass of a downwards rolling snowball). Also I came once across an article called "wexzal" where the authors use the inverse of $\ ^2x$ to solve for aeroplane propulsion, and for dynamics in the explosion "chamber" of a gun-shot.

I'll give another (own) example later where one looks at growth processes (like interest) over time and the result for one time-period is feeded back for a new time-period equivalent to the result.(I'll have to make it a bit more explicite).

For the justification of tetration of non-integer heights it might suffice to imagine a stack of replications of the complex plane. Mark a complex-coordinate $z_0$ at the lowest plane. Go one step higher and mark the complex number $z_1$ where $z_1 = b^{z_0}$ (where b is some base which you have chosen in the beginning) in the second plane. Proceed with iterating that operation for several (or infinitely many) steps.

Now we can try to imagine a vertical curve through all that points $z_0,z_1,z_2,...$ - in the simplest way of interpolation the vertical curve has a spiral form with windings around a fixpoint (if an attracting complex fixpoint exists for our base b and reachable from your initial value $z_0$). Surely there are infinitely many such curves, but it does not seem illogical that one of such curves has the most meaningful interpretation as interpolation for noninteger, real heights of our operation.
The imagination might possibly be improved, if we do not assume a staple of complex planes but having the planes rolled to cyclinders because of the periodicity of the exp-function for multiples of $ 2 \pi i $ - and possibly it is even better to imagine concentric spheres, whose surfaces represent the complex Riemann-sphere, and imagine then the connecting curves as trajectories of the real-values, continuous heights of iteration through the surfaces of the spheres.

The article of D. Kouznetsov (1) mentioned above contains an approach for interpolation to such real and even complex heights.

(1) D.Kouznetsov. (2009). "Solutions of F(z+1)=\exp(F(z)) in the complex zplane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7

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