I believe that we don’t know that to be normal in every base, just base 10. But I could be wrong.
Whats the chance of the sequence: 020915080126011804 showing up in pi?
I believe that we don’t know that to be normal in every base, just base 10. But I could be wrong.
Keep looking, BioHazard, your name’s in there somewhere.
As I understand it, every sequence shows up in pi. Some people have asked if this can be used to compress messages - just tell someone where in the expression of pi your message is. However, it would be no good, because in general, the first time a sequence N shows up is later than the Nth digit. So the position indicator would be bigger than the original message itself.
That was nice of them - they left a message specially for us creatures with 10 fingers. While pi may be universal, and natural, the base 10 number system is arbitrary. Or did they encode a message for each possible number base? I know, it’s only a book 
Hehe, I just noticed that smilies still show even in a spoiler tag - and when you select the happy smiley it turns into a frown.
I hear you BioHazard.
Here’s a 1998 popular article on the Borwein-Plouffe Algorithm:
http://www.maa.org/mathland/mathtrek_3_2_98.html
A more technical discussion:
http://www-2.cs.cmu.edu/~adamchik/articles/pi/pi.htm
And another:
http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.html
It is possible to compute the nth decimal digit of pi but not with the same efficiency as Bailey-Borwin-Plouffe.
Off topic, but pi has now been calculated to more than 1 trillion decimal digits; actually, 1,241,100,000 digits.
http://www.maa.org/mathland/mathtrek_3_2_98.html
I think pi is random in the sense that it passes all known randomness tests.
One of the most widely used definitions of “random” is based on Kolomogorov complexity: The size of the program to generate the string. Since you can generate huge number of digits of Pi using a tiny program, Pi is most definitely not random using this test.
Virtually any Computer Scientist will describe Pi as very nonrandom.
But [symbol]p[/symbol] probably would pass statistical randomness tests, which is probably what Shalmanese had in mind.
Good point; As far as I recall it’s not specifically mentioned, but if they were advanced enough to actually bend the universe in such a way as to modify a constant, I think we could assume that they would have engineered a certain degree of universality in the message.
Some people already made the point that “patterned” does not mean “rational,” with the example 0.12345678910111213… Even a transcendental number can be patterned. For instance,
[symbol]S[/symbol][sub]k=1[/sub][sup]infinity[/sup] 10[sup]-(k!)[/sup] =0.1100010000000000000000010… is transcendental but has a visible pattern.
Pi, as far as anybody knows, is random based on any statistical test or visible pattern in the digits, though it can be computed by fairly simple programs.
This spoiler thing is really cool. Actually Im pretty sure that it was not base 10 that she was using as she searched through pi. I think it was base 12. Im not sure. So the proof mentioned in Contact was irrelevent because any sequence imaginable is in there? Terry pratchet wrote a book based around what Sagan was hinting at, a long time before Sagan did. The book called Strata.
A happy smiley when I select it turns into an evil smiley when I select it.
Base 11 actually. Although I hardly see why somebody knowing this particular fact would ruin their reading of the story.
Let’s say someone DID find a pattern to pi or some other universal “mystery” if you will… Most people say that person would be rich, but how? Who would pay him for that information? And how WOULD you get payed? Would you just walk up to Universities or companies and tell them you discovered ______ and they need to pay you to tell them?
>Let’s say someone DID find a pattern to pi
The New Yorker profiled the work of the Chudnovsky brothers, back in the early '90s IIRC, and the article stressed the wave-like patterns that were found when the repetitions of the digits were plotted graphically; much like waves of the oceans.
- The motive behind the OP may be: what if some sort of analysis allowed one to actually “decipher” the decimal expansion of Pi, in such a way that it was found to contain an endless number of messages, not only in all languages known but in all POSSIBLE languages, with a content that was, in crucial respects, mutually consistent? (Maybe this is the idea in the Sagan novel; I’m also reminded of a Star Trek TNG episode in which ancient aliens encoded a video greeting in the DNA structure of all galactic life-forms). If someone insists on an evidentiary “proof of the existence of God,” it’s hard to imagine a better one.
But that’s just a speculation as to the OP’s motive.
- This distribution of prime numbers along the number line is also said to be “random” by our high school math teachers. But it turns out to have what I think is termed “order of the second degree.” Imagine creating a string of beads: black beads for nonprimes, white beads for primes, each representing an integer and ordered from one to N (N has to be fairly large). Make the first bead the centerpoint, and wrap the string compactly about that point in an outward-going spiral–creating a disk of beads, so to speak. If you have good eyesight, you will now see that the white beads form a very obvious pattern, rather like a poppy. If I’m interpreting this correctly: you cannot predict where the next prime will be, but you can assign varying degrees of probability to its appearance.
My point being–I wonder if some version of the “bead test” would reveal a VISUAL form in the expansion of Pi. Would any of you folks care to perform this test and report the results here?
Now that my juices are flowing…start with the first digit of the expansion of Pi (“1”) and pretend that as you go to the right, you are actually going left, in terms of decimal values. Ie, an unending string that begins 14159265358979323…etc. Now perform the bead test on that number (the rule is that you never have a lesser number after a greater, and you never jump to, for instance, three-digit numbers until you have exhausted the two-digit ones per the preceding clause). So the first few are:
1
4
15
92
653
5897
What will the bead test show regarding the distribution of primes in this series? Anything visually interesting?
I really like that idea, Scott. Someone try it? IANAProgrammer but I do think it would be relativly easy to make a program to do that. It would need to use the pi decimal finding algorithm as a base.
As for my OP, I was thinking more along the lines of Sagan’s book, Contact, with the “evidence” being proof of something greater than us.
If [symbol]p[/symbol] is absolutely normal, then it all is in there at some point. If not, 1415 can be made to represent anything you like by means of a suitable decoder. 
**
Again, if [symbol]p[/symbol] is absolutely normal, no large-scale pattern will emerge. If not, who knows?