Growth

No, not the phony kind you read about in self-help books. I’m talking about fast-growing functions, faster than exponentials. I found a way to illustrate it, I think, for the average person while tinkering with a spread sheet today.

Take a look.

Actually before we take a look, just some very basic concepts that anyone who can do grade school math understands.

First you have counting 1, 2, 3, 4…

If you want to speed up counting, you can add.

Instead of counting from three to six, you can just add 3 + 3. Addition therefore is just repeated (iterated) counting.

But instead of adding 3 + 3 + 3 + 3 + 3 + 3, you can just do 3 * 6 because multiplication is iterated addition.

But instead of multiplying 3 * 3 * 3 * 3, you can just do 3 ^ 4 because exponentiation is iterated multiplication.

That’s where most people leave off in their education, and they feel perfectly content to live their lives only because they don’t know what they’re missing. Beyond exponentiation lies tetration, or iterated exponentiation.

Instead of 3 ^ 3 ^ 3 ^ 3, you can just do 3 ↑↑ 4 (a power-tower of threes, four high.)

You would say it “three to the three to the three to the three” in exponentiation terms, and “three arrow arrow three” in tetration terms. The number before the arrows determines the base and the number after them how high the tower goes.

Let’s calculate some easy ones.

2 ↑↑ 2 is a power tower of twos, two high. That is 2 ^ 2 = 4.

3 ↑↑ 2 is a power tower of threes two high. That is 3 ^ 3 = 27. Easy.

2 ↑↑ 3 is a power tower of twos three high. That is 2 ^ 2 ^ 2 which is 2 ^ 4 = 16.

3 ↑↑ 3 is a power tower of threes three high. That is 3 ^ 3 ^ 3 which is 3 ^ 27 = 7,625,597,484,987.

Wait, what happened? You just encountered a fast-growing function.

Now we can take a look at the spread sheet:

On the left (rows) we increase the base number, and on the top (columns) we increase the height of the towers. Remember, x ↑↑ y is a tower of exes y high.

You can see which variable drives the growth. When you increase the base (x), you get the really big numbers making the triangle. X ↑↑ 2 is just a power-tower of exes two high, or x to the power of itself. So it’s 2 ^ 2, then 3 ^ 3, 4 ^ 4, etc., all the way down. That gets pretty big when it’s, say, 100 ^ 100, but the spreadsheet can handle it.

Contrast that with what happens when we increase the y term, i.e., make taller power towers. Even using two as the base (row 1), we can only get to 2 ↑↑ 5 before the spreadsheet can’t handle it, and returns an error. Why?

Because 2 ^ 2 ^ 2 ^ 2 ^ 2 blows up. It’s 2 ^ 2 ^ 2 ^ 4, which is 2 ^ 2 ^ 16, which is 2 ^ 65,536 which kills the spread sheet. (Consider 2 ^ 256 is already 10 ^ 77, and you can see how crazy taking it to the 65 thousandth power would be.)

When we use three or four as our base, we can’t get past y = 3 without blowing up the sheet, and with five as the base, we can’t get past y = 2.

But take a look how big the base (x) can get before y = 2 blows up the sheet:

It’s at 144 ↑↑ 2 which is 144 ^ 144 that the spread sheet can no longer deal with it, and it returns an error.

You can see how much more slowly increasing the x (base number) in x ↑↑ y grows the output than when you increase the y (build a higher tower.)

But if you really want to see obscene growth, you increase the number of arrows.

That’s because x ↑↑↑ y means x ↑↑ x y times. So 3 ↑↑↑ 3 means 3 ↑↑ 3 ↑↑ 3.

And 3 ↑↑ 3 (as we saw above) is 7,625,597,484,987.

So 3 ↑↑↑ 3 means 3 ↑↑ 7,625,597,484,987.

That is, a power-tower of threes 7.6 trillion high.

There’s a great post by Tim Urban (worth reading in full that goes much deeper than I will here) that shows how bananas a power tower that high is. But suffice it to say that even 3 ^ 3 ^ 3 ^ 3 breaks the spread sheet, and you have 7,625,597,484,983 more threes to go. It’s a number that’s unfathomable, but I’ll try to give it a little context.

There are 10 ^ 18 atoms in a grain of salt and 10 ^ 80 atoms in the universe.

10 ^ 100 is a googol.

10 ^ googol is a googolplex (the number has more zeroes than atoms in the universe.)

3 ↑↑ 5, a power tower of threes five high, is already bigger than a googolplex. And we need to get the power tower to a height of 7.6 trillion.

Of course, we only increased the number of arrows from two to three.

You could go to 3 ↑↑↑↑ 3, the explanation for which I’ll leave to Tim Urban.

If you want to go farther, check out how you get to Graham’s number (3↑↑↑↑3) = G1

And G2 has G1 arrows. When you get to G64, that’s Graham’s number.

But Graham’s number is virtually zero compared to Tree(3), and it only gets crazier from there.

I wrote a post on why we should care about this too.