# What would produce tetration in physics?

In a parallel universe much of the physics equation have tetration in them. There are things such as tetrational growth as well as tetrational decay. $x^x$ and $x^{x^x}$ are also found in some physics equations. Super logs and super roots are also necessary for understanding some of the physics of this universe.

What would produce tetration in physics? How would a universe with tetration involved in physics be different from our universe?

• – Trang Oul Nov 25 '15 at 8:54
• "$x^x$ and $x^{x^x}$ are also found in some physics equations. Super logs and super roots are also necessary for understanding some of the physics of this universe." Where? I think it would help this question a lot if you included the examples of these things that you're already thinking in - it might inspire answerers. Personally, I can't think of any such examples and if someone asked if there were examples, I'd say "no" for similar reasons to the answers given. However, if you included the examples you're thinking of, that would change how I would answer your question. – Milo Brandt Nov 26 '15 at 0:37

# no

It seems unlikely. In familiar physics we have an "action principle", constants, and integration leading to powers of an initial dimension.

What would raising a thing to its own power (rather than a pure number) even mean?

The referenced Physics Question puts it in succent terms: it doesn't have a good meaning for non-integer quantities. If real numbers are not bad enough, consider that units must disappear. If you raise a length to its own power, you can't refer to the number in some units; you get the concept of length raised to the length power. Now we understand length to an integer power (area, volume) and more generally multiplication and division of units (miles per hour), but units to powers of other units? What's that even mean?

In short, the question doesn't make sense. I can't think of what kind of universe that would be because I don't know what your proposed equation describes.

# but...

Apparently, the premise of the question is predicated on a misunderstanding. The links and further surfing from the related comment noted by Trang are very interesting.

Now the basic tools in math have been worked out as being useful for dealing with things in the world. Combining things, scaling, and exponential growth are all ubiquitous, as is spreading out over more (natural number) degrees of freedom.

Getting more abstract, pure math doesn't see addition and multiplication as the one-and-only kind of thing, but has monoids, fields, rings and such and can work with entities that are nothing like the "numbers" we understand.

Having the familiar relationship between two operations called + and × is a field, and you might wonder what it means to introduce more operations such that the relationship continues.

But it doesn't work out in general. You can intuitively describe it with natural numbers, but that's about it.

# But we can count fundimental things

Now what if you did have natural numbers? For example we can count the number of electrons. Given a set of n fundamental objects sharing a relationship, what about them could grow as a self-power? If every item connected to every other, that would be n squared connections. What kind of relationship can give a power that's a value based on an outside thing, not just a small integer?

Hmm, recursion! The number of links is the order of n squared, and the number of linked-to items not counting itself is (n−1). So you need to apply another level of relationship to the links and repeat that n times.

Now if you start handwaving you might remind the reader that this is like the strong nuclear force: gluons are themselves charged, and thus the links between things bearing color charge become more charged items. You need to limit that by having the new charged link not interact with the parent charged item but only others: that gives the recursion with reducing n on each level.

That gets you an interaction strength or resulting charge of $n^n$ based on n charged fundamental particles in a bound system. It still can't cope with tetration but it might give some ideas.

The particles that form links between initial permanent particles also have properties, and unlike in our universe they are different for each nesting so they know which generation they are from and (as explained above) which primaries they arose from all the way up the tree.

# another idea: hyperbolic planes

I recall a video from Numberphile on You-Tube concerning rukes for sports played on a hyperbolic plane. The area growth is much greater, but I don't recall the order.

# logs

Trying to contemplate the power, I thought of how you use logs to find such a value. Given links between nodes you get squaring of the initial value as the number of such links. But processes such as sorting, ordering, and traversing will give rise to n log (n) relations. That's also how you raise a power: by multiplying logs. Comparing that with the algorithm for raising to a power, what's still needed is the final step of exponential on the previous result, and that is a natural kind of operation (growth curve).

So, you need an initial seed value (the sum of the initial n charges) undergoing generations of exponential growth keyed by the complexity of sorting those initial n items. That would be $e^{n\ ln (n)}$ which is the same as $n^n$ which makes this an approach that answers the original question.

# but would such a universe be interesting.

If forces etc. grow with such kinds of curves, you won't have a selection of things in the same range. Rather than rich combinations of interactions like we have with molecules, one thing will utterly trump everything else.

But the Wikipedia page shows some very complicated graphs: fractals formed from graphs involving this type of function.

So what if these fractals describe the physics? Not charges, space, time, movement, and forces as with our universe, but some means of time evolution of state based on changing parameters on such a curve, where the "field" is explained by this kind of function.

The problem is having a universe too alien for an understandable story.

So look at where fractals pop up in real life: chaos. The initial discovery was with weather. The scale of the beings describing their universe might be ruled by fluid dynamics or something like that with emergent properties, and taming that involves understanding fractals created by iterating functions.

• In order words: such a universe could not be flat, thus it would curl up into itself and have zero volume. i.e. not exist at all. ;) ...unless its volume is $0^0$! ;o – Draco18s no longer trusts SE Nov 25 '15 at 23:13
• I thought of something, and updated the post. – JDługosz Nov 25 '15 at 23:47
• A hyperbolic universe probably couldn't support life. It would be Interesting, but only in a theoretical sense. I'm not entire sure solid matter could occur in such a universe (due to the "shortest distance" problem). My last comment was more of a joke on the $x^x$ when x = 0 having a humorous implication. – Draco18s no longer trusts SE Nov 26 '15 at 5:31
• I think a universe needs a concept of time to have the state change over time, but doesn't need space. I used the idea of financial bookkeeping to muse over the concept: rules and time but the state can be anything. You might have nodes with connectivity describing the allowed interactions, but not space or motion. – JDługosz Nov 26 '15 at 6:10
• True, but it won't be very interesting. As no large-scale structures could form, you'd be stuck watching an endless soup of sub-atomic particles swirling about. – Draco18s no longer trusts SE Nov 26 '15 at 14:44