The Physical Meaning of Tetration with fractional power tower 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 $^{2.5}2$ ?
 A: 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)
