May 26, 2014 0

## How big is africa?

I know I’ve seen this before, but was just discussing with my mother and feel like it’s worth posting in case someone hasn’t: Africa is larger than China, the US, India, and all of Europe (combined).…

May 26, 2014 0

I know I’ve seen this before, but was just discussing with my mother and feel like it’s worth posting in case someone hasn’t: Africa is larger than China, the US, India, and all of Europe (combined).…

May 19, 2014 1

…are what we strive for in science. No really: check this out.

One of the things that he doesn’t explicitly say, but which I think fits in exactly the same vein, is as follows. There is no algorithm to do research. You can’t say “It will take me three days to come up with a solution for that problem.” The whole point is that *nobody knows* how to solve the problem. That’s what makes it awesome.…

May 17, 2014 0

How many seconds would it take a monkey, randomly hitting keys on a typewriter, to spell out Shakespeare’s complete works? That number is tiny compared to what we’re going to discuss today! I’ll start with a quote:

There is a diamond mountain a thousand cubits tall. Once in a hundred years the fairy Lilavati comes dancing through the air and brushes the top of the mountain with the end of her train. When the whole mountain has been worn away, the first instant of eternity will have passed.

Unfortunately I’m no longer sure where the quote is from, although the beauty remains in any case. I thought I had originally found it in Anglin’s history of mathematics, but both there and on the web could only find reference to the great 12th-century indian mathematician Bhāskarāchārya and his beloved daughter Lilavati. So perhaps that is the correct connection, but I’m not sure about the history of the tale. I did find a reference to a story by the Brothers Grimm about a shepherd boy (*Das Hirtenbüblein*) in which a king asks the boy how many seconds there are in eternity. He answers:

The Diamond Mountain is in Lower Pomerania, and it takes an hour to climb it, an hour to go around it, and an hour to go down into it. Every hundred years a little bird comes and sharpens its beak on it, and when the entire mountain is chiseled away, the first second of eternity will have passed.

I like the fairy version better, but clearly they have the same origin. The reason I started looking back into all this was that I serendipitously came across Robert Munafo’s wonderful website about large numbers. Most of the site involves going through a sequence of constructions of large but finite numbers, roughly in order of their magnitude. For instance a googolplex (10 to the power of 10^100) is the largest number that most people have heard of (and that has a well-used name). It’s down the list, but not very far down the list.

The ones that I remember from math camp when I was a kid (although I didn’t know the name) are the Steinhaus-Moser numbers. *n* “in a triangle” is *n*^*n*, for positive integers *n*. So 5 in a triangle is 5^5 = 3125. Interestingly Munafo says he learned about these in 1979 – my math camp was in the mid 1980s, so apparently I was pretty up to date at that point!

To continue our story: *n* “in a square” is *n* in *n* triangles (i.e. *n* in a triangle, itself in a triangle, itself in a triangle, … *n* times). And so on. 2 in a pentagon is called mega, which is already way beyond the magnitude needed to describe anything in the physical world, even the number of possible configurations for all the atoms in the known universe. 2 in a mega-gon (i.e. a regular polygon with number of sides equal to mega) is called moser; this is frighteningly large.

But this is not the end of the line by any means, although it starts getting more and more impressive to even describe the staggeringly huge numbers that result, and more and more difficult to compare them to one another. The last sets of finite numbers that he describes come from computability theory and involve the “busy beaver” function, which is another wonderful image, and ultimately oracle turing machines. I was not familiar with very many of these at all, and it was a lot of fun …