Mathematics: Can +, x and ^ operands be generalized to higher orders?

Ok so here is one mathematical oddity that has me puzzled for some time, well 3 years to be precise. Yesterday I finally decided to take a pen and a paper and see if I could make any headway to put myself to sleep, here is what I wrote down.

Definition of Operands at degree n: OP(n)

OP(0): a OP(0) b == a + b

a OP(n) b == a OP(n-1) a OP(n-1) ... OP(n-1) a (b times)

you then have

  1. OP(0): by definition it is +

    Operand zero is the sum (by definition)

  2. OP(1): a OP(1) B = a OP(0) a OP(0)... OP(0) a (b times) = a + a ... + a ( b times) = a x b , which implies OP(1) == x

    Operand one is the multiplication

  3. OP(2): a OP(2) B = a OP(1) a OP(1) ... OP(1) a (b times) = a x a ... x a (b times) = a ^ b, which imples OP(2) == ^

    Operand two is the power operation

So in other words with this definition of operands we can trivially rebuild the 3 familiar operands (+, x , ^).

Let's go on and look at Operand 3:

OP(3): a OP(3) b = a OP(2) a OP(2) ... OP(2) a (b times) = a ^ a ^ ....^ a (b times)

for example a OP(3) 2 = a ^ a.

3 OP(0) 3 = 3 + 3 = 6
3 OP(1) 3 = 3 + 3 + 3 = 3x3 = 9
3 OP(2) 3 = 3 x 3 x 3 = 3 ^ 3 = 27
3 OP(3) 3 = 3 ^ 3 ^ 3 = 3 ^ 27= 7.6 e 12

LOL what a weird bird. 3 OP(3) 3 = 8 e12

and a OP(3) n = a power a power a ... power a (n times) which kind of boggles the mind. I will call OP(3) bogglemind :)

I have tried to compute its derivative, needless to say I haven't made much progress but I did compute the following result (likely completely false):

d((x OP(3) n))/dx = (x OP(3) n) (x OP(3) n-1)/ x

It says that its derivative is roughly itself squared..... You will excuse the lack of rigorous presentation in my dissertation:) At that point, I fell asleep counting so many sheep.

It could be easy to program the various OP(n), however OP(3) already puts associates the duplet 3,3 to e12 so.... it will blow computing power very quickly. Analytical means may be the only recourse.

But the question that is really really puzzling me since the first time I thought about OP(3) is that Nature makes extensive use of OP(0), OP(1), OP(2). Why does it stop there? is there any place where OP(3) could emerge?

Nature makes extensive use of OP(0) (the sum), OP(1) (the multiplication) and you likely use them in your everyday life.

Even OP(2) is found everywhere although you may use it less in everyday life: in physics with electromagnetism, in finance with compounding interest, in biology with growth of compounds in transcription networks, in chemistry with reaction dynamics, and even in theory of knowledge, see Kurzweil for example and the "singularity": your new knowledge grows proportional to your existing knowledge (you build new knowledge on your existing knowledge) which is the very definition of exponential growth over time.

But what about OP(3)? why hasn't this cuddly monster raised its pretty head? Surely a phenomena that puts integers 3 and 3 in relation with e12, is a weird weird physical phenomena. The domains I have encountered that span of 10 orders or magnitude in a given finite time frame have been:

  1. Biology: cell metabolism. A few compounds can influence a cell counting e10 proteins.
  2. Weather: Butterfly effects.
  3. Physics: Renormalization theories, my master thesis was on renormalization of non-abelian gauge theories. Don't worry I don't remember what it was about either. However I remember this: when parts of the equations blew up, theoretical physicists would take the very pedestrian approach of "taking them out". Which always felt like a hack to me. I have read since that there was a geometrical basis for such a surgical removal of the "blow ups" but maybe the blow up have a real causal basis?
  4. Information Theory in the age of the Internet: can a few good men really influence the world population in a finite amount of time? 1 -> 5 e 9. It relates to the critical mass of information on the Internet (JBoss's visibility from 0 to 60 in 2 years)

But a real avenue of exploration may be: phenomenas whose derivative is the square of themselves. dF/dx= F^2.

Things that grow as the square of themselves would make use of OP(3). Can someone name physical phenomenas? I have no idea very frankly, I just don't remember. I just find it amusing and entertaining.

If OP(0,1,2) are so popular in Nature, why would OP(3) have no role? Maybe we just live in, observe and try to explain a world to our proportions and the proportions of 1->e12 span our current existence and physical reality. I hope a real mathematician has already looked at this problem. I would love to hear from any of the 1000 readers/week if any of you has heard of such a thing and has a link. I am convinced OP(3) has been talked about already.


Juha Lindfors said…
If you can count to 7.6 e 12 sheep and still not fall asleep, you will break the mainframe.

That's it.
Roy Russo said…
d((x OP(3) n))/dx = (x OP(3) n) (x OP(3) n-1)/ x = 42

There. Now go to sleep.
Julien Viet said…
Funny I have been wondering about the same idea for many years.
Barry Kelly said…
At first glance, OP(n) is equivalent to Knuth's up-arrow notation for n-1 up-arrows, where n >= 2. Iterated exponentiation, your OP(3), is called tetration, and has its own format - a bit like exponentiation in the mirror.
Unknown said…
I knew there would be jokes about 42 and mainframes :)

I feel slightly old and very crazy.

Come on! there has got to be something somewhere.
Unknown said…
barry! Bingo!

As you can see, I was writing my answer while you were writing yours.

Yes! this is exactly it. The OP(n) defined here are the Hyper-Operator or the Knuth up arrow notation! Apparently hypothesized in the 76 by Knuth. The OP(3) is called "Tetration or Hyper 4".

I browsed what I could from wikipidia and found the material a bit weak. No discussion of derivatives nor discussions of APPLICATION.

Tetration is probably FOUND in phenomenas that have their derivative equal to the square of themselves.

Very cool barry, thank you, how did you go about finding this information? What did you google for? did you know about it before? from knuth? very cool, thanks
Unknown said…
well, :)

by googling tetration :)

There is good stuff on the physics (indeed found in renormalization).
It has to do with Hopf algebras and 90 pages of stuff I didn't understand at all. But if you want to take a look at what experts are doing in theoretical physics and all the cute little diagrams that look like fishes. I wonder if what we do in software looks dumb in comparison?

Don't waste more than 30 seconds, it is completely obscure.

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