When you repeat addition you get multiplication. When you repeat multiplication you get exponentiation. And when you repeat that, you get

Let's follow Donald Knuth and write exponentiation as ↑. Then 4 cubed is

4↑3 = 4 × (4 × 4) = 64

We are multiplying 4 by itself 3 times. Next comes tetration, which we write as ↑↑. Here's how it works:

4↑↑3 = 4 ↑ (4 ↑ 4) = 4 ↑ 256 ≈ 1.3 × 10¹⁵⁴

We are raising 4 to itself 3 times. Before the parentheses didn't matter, but now they do: we put the parentheses all the way to the right, since (a ↑ b) ↑ c equals a ↑ (b × c) while a ↑ (b ↑ c) is something really new.

As you can see, tetration lets us describe quite large numbers. 10↑↑10 is much bigger than the number of atoms in the observable universe.

In fact, 10↑↑10 is so big that you couldn't write it down if you wrote one decimal digit on each atom in the observable universe!

In fact, 10↑↑10 is so big you couldn't write the

In fact, 10↑↑10 is so big you couldn't write down the

We could march forwards and create notations for numbers so huge that 10↑↑10 looks pathetically small.

This makes no sense at first: you can't write down a tower of powers with 10.2 tens in it. But you couldn't add 10.2 tens together at first, either, so 10 × 10.2 didn't make sense until someone explained what it meant. Same for exponentiation. So, maybe tetration can also be defined for fractions in some nice way. And maybe real numbers too. And maybe even complex numbers.

Believe it or not, this seems to be an open question! It's phrased as a precise conjecture here:

http://en.wikipedia.org/wiki/Tetration#Extension_to_complex_heights

and there's a good candidate for the answer. The picture here, drawn by Dmitrii Kouznetsov, shows a graph of the candidate for e↑↑z as a function on the complex plane.

#bigness

**tetration**!Let's follow Donald Knuth and write exponentiation as ↑. Then 4 cubed is

4↑3 = 4 × (4 × 4) = 64

We are multiplying 4 by itself 3 times. Next comes tetration, which we write as ↑↑. Here's how it works:

4↑↑3 = 4 ↑ (4 ↑ 4) = 4 ↑ 256 ≈ 1.3 × 10¹⁵⁴

We are raising 4 to itself 3 times. Before the parentheses didn't matter, but now they do: we put the parentheses all the way to the right, since (a ↑ b) ↑ c equals a ↑ (b × c) while a ↑ (b ↑ c) is something really new.

As you can see, tetration lets us describe quite large numbers. 10↑↑10 is much bigger than the number of atoms in the observable universe.

In fact, 10↑↑10 is so big that you couldn't write it down if you wrote one decimal digit on each atom in the observable universe!

In fact, 10↑↑10 is so big you couldn't write the

*number of its digits*if you wrote one digit of*that*number on each atom in the observable universe!!In fact, 10↑↑10 is so big you couldn't write down the

*number of digits in its number of digits*if you wrote one digit of*that*number on each atom in the observable universe!!!**Puzzle:**About how many times could I keep going on here? Let's say there are 10 ↑ 80 atoms in the observable universe; this seems roughly right.We could march forwards and create notations for numbers so huge that 10↑↑10 looks pathetically small.

*And we will!*We can even look at infinite tetration.*And we will - that's our main goal here!*But it's also fun to ask questions like:**Puzzle:**What is 10↑↑10.2 ?This makes no sense at first: you can't write down a tower of powers with 10.2 tens in it. But you couldn't add 10.2 tens together at first, either, so 10 × 10.2 didn't make sense until someone explained what it meant. Same for exponentiation. So, maybe tetration can also be defined for fractions in some nice way. And maybe real numbers too. And maybe even complex numbers.

Believe it or not, this seems to be an open question! It's phrased as a precise conjecture here:

http://en.wikipedia.org/wiki/Tetration#Extension_to_complex_heights

and there's a good candidate for the answer. The picture here, drawn by Dmitrii Kouznetsov, shows a graph of the candidate for e↑↑z as a function on the complex plane.

#bigness

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- +Romain Brasselet - the Wikipedia article is pretty helpful if you're trying to figure out what 10↑↑10.2 is:

http://en.wikipedia.org/wiki/Tetration#Extension_to_real_heights

It says there's no generally agreed-on answer, but it has a lot of useful facts and references, and it presents a candidate that seems good to me. When it gets to complex heights, it gives a link to Mathematica code that calculates what*may be*the unique best version of x↑↑z when z is complex. I haven't had the energy to make any progress on these questions myself....Nov 27, 2012 - >the nod of 10↑↑n is 10↑↑(n-1) +1

Or, the log of 10↑↑n is 10↑↑(n-1), period, which is slightly easier to calculate with.

Here, by ‘log’ I mean the floor of the common logarithm, which is always 1 less than the number of decimal digits of a positive integer.Nov 27, 2012 - I've been looking at the for a week now and I still have no idea how to read it... but its bloody beautifulDec 6, 2012
- +John Baez - As the author of http://www.tetration.org I have been deeply involved in extending tetration to the complex numbers as well as matrices. You may be interested to know that Philippe Flajolet's research and your writing have had the biggest impact on my research since the turn of the millennia. Back in 2006 I noticed an article while searching on “tetration” with Google - it was the Wikipedia article on tetration. I made what I felt were significant improvements to the article, but after struggling for a year with people who didn’t follow Wikipedia’s policy of no unpublished personal research I didn’t want my name associated with the article.

Here are some things to consider regarding extending tetration. For links on who has done what regarding tetration see http://tetration.org/Links/index.html.

Extending tetration is just an example of a much broader and important problem, that of fractional iteration. If you conquer fractional iteration you can not only extend tetration, but also pentation and all the higher operators of the Ackermann function. I met with Stephen Wolfram in 1986 several times and discussed my work with extending tetration. Wolfram was very interested because at the time he was trying to unify different dynamical systems into a single system that could mathematically model any dynamical system. There are two dynamical systems deemed capable of representing the dynamical systems in physics - PDEs and iterated functions. The problem is that iterated functions are discrete while physics given every appearance of being continuous. Therefore a theory of fractional iteration would provide a new inroad into understanding physics. See

R. Aldrovandi and L. P. Freitas,

Continuous iteration of dynamical maps,

J. Math. Phys. 39, 5324 (1998)

for a great paper on continuous/fractional iteration with an application to the Naiver Stokes equation. My derivation of fractionally iterated functions is experimentally consistent with Aldrovandi and Freitas, but I haven’t proven their mathematical equivalence.

What combinatorial structure is associated with iterated functions. Look at a simpler problem and ask what combinatorial structure is associated with composite functions? The answer is set partitions or Bell numbers. Aldrovandi and Freitas paper follows this line of thought and use Bell matrices in their paper. Riordan’s book Combinatorial Identities and the chapter on Bell polynomials gives further insights. Bell polynomials provide a way of representing arbitrary functions in terms of the derivitives of composite functions. The combinatorial structure associated with Bell polynomials has a number of names: Schroeder’s Fourth Problem, hierarchies total partitions and phylogenetic trees. See http://oeis.org/A000311 for more information. My own work focuses on the derivatives of iterated functions which are Bell polynomials because iterated functions are trivially composite functions. See http://tetration.org/Combinatorics/index.html to view how Schroeder’s Fourth Problem is recovered from the coefficients of the derivatives of iterated functions.

Tetration is based on iterated functions which are based on complex dynamics. Working alone for decades it has become very important for me to find flaws in my extension of tetration. The research community mostly consists of I. N. Galidakis, and myself working as individuals and researchers associated with the Tetration Forum including Kouznetsov http://math.eretrandre.org/tetrationforum/ . My point of difference with the Tetration Forum and Kouznetsov is that they are proposing solutions to the extension of tetration that are easily seen as being inconsistent with complex dynamics. In fact they claim that fixed points are irrelevant to extending tetration. According to complex dynamics -

Lennart Carleson, Theodore W. Gamelin

Complex Dynamics

Springer-Verlag

In reviewing Chapter 2 - the Classification of Fixed Points, one can see that because of the conjugacy of fixed points that there can be no “simple” single equation to extending tetration. In the classification of fixed points there are hyperbolic, super-attracting, parabolic rationally neutral, rationally neutral and irrationally neutral fixed points. Each type of fixed point leads to a different type of tetration. For example e ↑ (1/e) is the sole parabolic rationally neutral fixed point for tetration, so e ↑ (1/e) must have it’s own separate equation for extending tetration. Not only is my work consistent with the Classification of Fixed Points, it provides an algebraic explanation of the Classification of Fixed Points that is much simpler than the topologically based proofs in Complex Dynamics.

While I have been able to fractionally iterate any complex function and therefore tetration, there is a major problem everyone researching tetration faces, that of convergence. In my mind all the real action in tetration is focused on people showing that they have a convergent solution. A couple of years ago the issue was quite active on MathOverflow. See http://mathoverflow.net/questions/tagged/fractional-iteration .Mar 8, 2013 - I think it'd be better if the "correct" ordering of brackets shown above was considered just one case and if the theory introduced a new concept similar to commutativity, except for tetration, governing in which order the powers are applied. For example Googol = (10↑10)↑10 whereas 10↑(10↑10) is ten times that. In "super tetration" any tetration would require a set with modulus equal to the 2nd term of the tetration minus 1, governing the order in which powers are applied. I imagine that would be much more versatile. If you really wanted to make it interesting you could then ask what if the "ordering" set contained non-integers, you could see what happens if the two (or more) powers are applied at the same time instead of sequentially! Actually doing it... that's something for a better man than me!Sep 17, 2014
- +Romain Brasselet We can reach 9775.29297749 likes yet.19w

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