All of Life in Pi?

Published in IEEE Spectrum Magazine, July 2013

Many years ago I was having dinner with an eminent mathematician, and in the course of conversation I mentioned an episode regarding the number pi in Carl Sagan's book, Contact.
A scientist from earth (Jody Foster in the movie) visits the distant planet and learns that the aliens have calculated pi out to many places, and that very far out in the expansion the decimal digits turn to a very long string of ones and zeros -- a message embedded in pi.

"How can you hide a message inside pi? It's built into the fabric of the universe," asks the scientist.  "And what does the message say?"

"We don't know.  We're still working on it," replies the alien.

This episode had made an impression on me.  What if there were a message in pi?  It is an almost mystical number, appearing in Einstein's theory of general relativity and in many other fundamental equations that describe physical reality.

But my dinner companion only stopped chewing for a moment.  "Well, of course," he said.  "It's an irrational number and contains all possible sequences."
"Well, dah," I thought.  Feeling stupid, I changed the topic of conversation.

I had the occasion to remember this exchange recently when there was an outpouring of discussion on Facebook following the posting of a photo showing several thousand digits of pi in a faint background and in the foreground the following text:

"Pi is an infinite, non-repeating decimal -- meaning that every possible number combination exists somewhere in pi.  Somewhere in that infinite string of digits is the name of every person you will love, the date, time, and manner of your death, and the answers to all the great questions of the universe."

From a sampling of the thousands of comments and blogs that were stimulated by this posting, I was made aware of one surprising fact that I wished I had known during that dinner so long ago -- that it is not true that because pi is an irrational number-- a non-repeating infinite decimal -- it therefore contains all possible number combinations.  Oh, while it almost certainly does, this has never been proven mathematically.  There are examples of irrational numbers that do not contain all possible sequences.  The so-called "normal" numbers do contain all sequences in the same statistical frequencies as true random numbers, but it is not known if pi is normal.  However, it has passed tests for statistical randomness out to many places, and since virtually all numbers are normal anyway, we can assume pi is also.

But is all of life written in pi?  No.  There is nothing there.  For every fact you might find, you would also find the exact opposite.  For every name of someone you might love, there would be countless other names.  Shannon would tell us that a sequence of random numbers contains no information.  He describes information as "the removal of uncertainty."  No uncertainty is removed through any perusal of the digits of pi. Moreover, the finding of any specific text is so statistically improbable that I would move it into the impossible category, like the old story about monkeys typing Shakespeare.  To simply find your own name, for example, might require 15 ASCII characters, which would equate to 36 decimal digits.  I haven't worked out the number of digits you would have to search to have a reasonable probability of finding a run of length 36 correct decimals, but I believe it would be far more than the current record for expansion of pi, which is 10^13 digits.

Still, maybe if we expanded pi out a few more places and a message appeared -- that would be scary.  Something to think about.