On the (Small) Number of Atoms in the Universe

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.

On the number of Go positions

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 40-character password:

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

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. 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:

12,000,000 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? 10^86696638 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

• Combinations of things are multiplicative, while collections of things are additive. If you see a line of 6 people, it is easy to visualize a line of 60 people—it is ten times longer. But even if you know that there are 720 different orderings (permutations) in which those 6 people can line up, there is no way you can visualize the number of orderings for 60 people, because it is—you guessed it—larger than the number of atoms in the universe.
• Big numbers are hard. Even with simple collections of things, it takes practice to get a real intuition for the difference between 6 million and 6 billion people. The number of combinations of things grows much faster and therefore intuition fails earlier.