Fun Fact - Messages in π

Have you read Contact, by Carl Sagan, or seen the movie? The movie is very true to the book, which doesn't always happen; but mercifully, the movie left out a rather silly passage at the end of the book, wherein they found a mysterious message within the digits of π, perhaps a message from God. My immediate reaction was: there can be no message in π. No one can put a message in π, not even God, because it is the result of a taylor series from a differential equation, and the digits spill out as they will, in any universe, under any laws of physics, or even in no universe at all. You could hide a message under a rock, but not in π.

Then I thought about it some more and I realized, of course that message is in π. In fact, every message is in π. The premise, which is born out by examining billions of digits, is that the digits of π look essentially random. So treat them that way. If you're looking for "hello world", you need 22 digits in a row that are the ascii codes for these 11 letters. The odds of that happening at random is 10 to the 22. That's a big number but π is infinite, so after you've examined the first 1022 digits you'll probably find "hello world". If you want a copy of War and Peace, you'll find it using the same reasoning. Sure you have to look longer, but it's there, in fact it has to be there! As you look farther and farther down the digits of π, the odds of finding War and Peace, exactly as Tolstoy wrote it, approach certainty. Every book, every message, every thought, is in π, or any other transcendental number for that matter. That's a bit mind blowing in a way, but it also has no practical value.

I suppose Contact was claiming this message was found implausibly early, i.e. within the first billion digits of π. Yes, that would cause you to take a step back. But again I say, it can't happen. The worm holes zipping people around the galaxy at flt speeds didn't bother me, but that little detail did. Thank you James V. Hart for leaving it out.

Isolated Bits

Here's a little known yet fascinating fact about π. You can compute the nth digit of π, without computing all the prior digits, but only in base 2. Thus you could determine the trillionth bit, even though we don't have the resources to compute the first trillion bits. I've never seen this theorem, and don't know how it works, I only know it exists.

Rebase

If an irrational number looks random in one base, does it look random in every base? I don't think this is assured. Build an irrational number r from an infinite string of random digits, but whenever 1 and 2 appear in sequence, make the next digit a 3. That destroys randomness; 3 always follows 12. But switch to base 7 and I bet it looks random again.

Further Reading

Contact the movie.