I’ve heard that because pi is a transcendental number, there is no discernable pattern in its digits, and that this means that ANY sequence of digits can be found somewhere in pi. If this was true, you should theoretically be able to create a binary representation of pi and find any computer program, mp3 file, or DVD image in it you wanted. Just take an mp3 file and keep expanding pi until the sequence of digits matches the file. Then, you could send the file to people just by telling them which digits of pi to begin and end with. Is this actually possible?
I think you’d need an infinitely large computer memory, running for an infinite time. But other than that, I don’t see any problems!
We don’t know if it works with pi. But any normal number will do; just pick one 
Sure it’s possible, see http://www.facade.com/legacy/amiinpi/. Other people with way too much time on their hands have looked for patterns in pi.
What we know: First, there’s numbers which are called “normal” which have this property. Second, there’s far more normal numbers than there are non-normal numbers, although specific examples are hard to come by. Third, it’s very widely believed (though not yet proven) that the digits of pi are normal. Fourth, you could encode information in this way, but on average, it’d be a lousy way to do it: The number representing the starting point would be bigger than the original file, and it’d take an extremely long time to encode or decode anything that way.
Even if examples of normal numbers are “hard to come by”, is there even ONE number known to mathematics that can be proven normal? And, if so, how can such a thing be proven?
First off, a distinction: an absolutely normal number has the property that any sequence can be found in its digits in any base. There are numbers known to be normal in base 10 (think .1234567891011121314…), but if pi were proven to be absolutely normal, then it would be the first known absolutely normal number.
As for how one goes about proving these things…well, I can’t answer that.
When you show me a computer program that can do that, I’ll show you my shakespeare typing monkey. I mean this in all seriousness.
Better have your Shakespeare-typing monkey ready, cause if pi is normal, I can do that. For you see, there are already programs that will calculate the nth digit of pi without having to know any of the others.
Even if that is so ultrafilter, your PC will have a massive task to do that for every single byte of The Matrix DVD in order to generate the file.
The vastly bigger task is calculating pi sufficiently to find matches for specific files.
[exhausted computer scientist]Okay, we pooled all the computer resources of earth and of all nearby galaxies and finally found a sequence in pi that matches Steely Dan’s Aja CD except that the lyric to Deacon Blues goes "learn to work the sousaphone and drink ovaltine all night long and When Jose Comes Home is an alternate demo version. That will have to do.
While it’s light years from being practical I think it’s one of the more interesting thought experiments I’ve heard in a long time.
Never said it was feasible, just possible.
I used tidyman’s link to find where the pattern “012345” appears in Pi. It’s at 447856. So instead of telling you “012345” I could tell you “447856” - which didn’t reduce the amount of data at all! I think the size of data and the number of digits to describe its location in Pi will always be of the same order of magnitude.
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*Originally posted by ultrafilter *
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This reminds me of a quote some years back on the average intelligence of the human race and the Internet:
'It used to be said that a team of monkeys with typewriters, given enough time, could recreate the works of Shakespeare.
Thanks to Usenet, we now know that this is not true’
After rolling this idea around in my head a bit, it occured to me. Not only is all currently known data encoded somewhere in the depths of PI, but all data that will ever be created is in there too.
Imagine it, somewhere in that infinite tangle of digits is Baywatch 2005, A few dozen StarTrek sequels and tomorrows copy of the Times.
All we need now is a Seer to divine this wisdom and dole it out Nostradamus style.
There’s a wonderful cyberpunk short story that muses on this theme, “Pi in the Sky” by Rudy Rucker. Aliens send a message on how to decode the data in Pi.
Carl Sagan uses this in the book Contact, but it’s not in the film.
Also not entirely unconnected to this topic is the story Library of Babel (I’ve seen modern rewrites of this concept too).
But we will only know what it is once we’ve seen it (not unlike Nostradamus actually, where the prophecies can only be successfully applied in retrospect)
Not only that, but every possible variation on tomorrow’s copy of the Times is in there too. Blows your mind if you think about it too hard.
I remember that. The first homework assignment in my second discrete math class was to read that, figure out how many distinct books there were, devise an algorithm to determine whether all the books were unique, and to find three words we didn’t know. That was an interesting class.