On the (Small) Number of Atoms in the Universe

The number of atoms in the observable universe is about 10 to the 80th power (1080). This measure is frequently used as a canonical really big number. There certainly are a lot of atoms in the universe. As a leading expert said,
Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.
But sometimes "number of atoms in the universe" gets used in contexts where it is an inappropriately small number.

On the number of Go positions

While discussing the complexity of the game of Go, Demis Hassabis said:
There are more possible Go positions than there are atoms in the universe.

A Go board has 19 × 19 points, each of which can be empty or occupied by black or white, so there are 3(19 × 19) 10172 possible board positions, but "only" about 10170 of those positions are legal. Here are some of them:

The crucial idea is, that as a number of physical things, 1080 is a really big number. But as a number of combinations of things, 1080 is a rather small number. It doesn't take a universe of stuff to get up to 1080 combinations; we can get there with, for example, a passphrase field that is 40 characters long:

a correct horse battery staple troubador

A reduced 13 × 13 Go board also has about as many positions as the number of atoms in the universe; the full 19 × 19 board has 1090 times more. So saying that there are more Go positions than the number of atoms in the universe is a bigger understatement than saying the national debt is more than a penny. (I'm sure Demis is acutely aware of this, and used "the number of atoms in the universe" for rhetorical effect, not for accuracy.)

On the number of digital pictures

Let's switch from Go positions to digital pictures. There is an art project to display every possible picture. Surely that would take a long time, because there must be many possible pictures. But how many?

We will assume the color model known as True Color, in which each pixel can be one of 224 17 million distinct colors. The digital camera shown below left has 12 million pixels, and we'll also consider much smaller pictures: the array below middle, with 300 pixels, and the array below right with just 12 pixels; shown are some of the possible pictures:

      
      
12M pixels 300 pixels 12 pixels

Quiz: Which of these produces a number of pictures similar to the number of atoms in the universe?

Answer: An array of n pixels produces (17 million)n different pictures. (17 million)12 1086, so the tiny 12-pixel array produces a million times more pictures than the number of atoms in the universe!

How about the 300 pixel array? It can produce 102167 pictures. You may think the number of atoms in the universe is big, but that's just peanuts to the number of pictures in a 300-pixel array. And 12M pixels? 1086696638 pictures. Fuggedaboutit!

So the number of possible pictures is really, really, really big. And the number of atoms in the universe is looking relatively small, at least as a number of combinations.

On counting combinations

People often underestimate the number of combinations of things. I think there are two main reasons: So beware, and be sure to use some simple math to augment your intuition when dealing with combinations.


Peter Norvig — 20 April 2016