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In Carl Sagan's novel Contact, the main character (Ellie Arroway) is told by an alien that certain megastructures in the universe were created by an unknown advanced intelligence that left messages embedded inside transcendental numbers. To check this, Arroway writes a program that computes the digits of $\pi$ in several bases, and eventually finds that the base 11 representation of $\pi$ contains a sequence of ones and zeros that, when properly aligned on a page, produce a circular pattern. She takes this as an indication that there is a higher intelligence that imbues meaning in the universe and yaddayaddayadda.

I always thought that Sagan was pulling a fast one on us. Given that transcendental numbers (or, for that matter, irrational numbers) are infinite, non-repeating sequences of digits, they contain any possible sequence of numbers, including the one Arroway found. It's hard for me to infer anything philosophical/spiritual from the fact that you can find this sequence if you look hard enough (to make a somewhat facile comparison, if I look hard enough in my sock drawer, I will find both socks of any given pair, but you can't take this as evidence for a higher intelligence in the universe). But then, Sagan did know one or two things about math, so maybe I'm missing something here. Are there any circumstances in which finding a particular sequence in a certain position of $\pi$ would make mathematicians go "wow!"?

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    $\begingroup$ It's not known whether $\pi$ contains every possible sequence of numbers. A sufficient condition for this to be true would be if $\pi$ is a normal number: en.wikipedia.org/wiki/Normal_number This seems likely to be the case since almost every real number is normal (the set of non-normal numbers has measure zero). But this has not been proven. See also this question: math.stackexchange.com/questions/216343/… $\endgroup$ – Bungo Jan 14 '15 at 22:17
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    $\begingroup$ Tragically, the aliens prefer $\tau$ and we missed their message. $\endgroup$ – user4894 Jan 14 '15 at 22:19
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    $\begingroup$ Not related to the math side of things, isn't Contact sort of about faith? The question of "is this sequence significant, or is it pareidolia?" continues that theme. $\endgroup$ – KSmarts Jan 14 '15 at 22:21
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    $\begingroup$ Also, as @Bungo points out, not every transcendental number contains every possible sequence of numbers. For example, Liousville's Constant, the first proven example of a transcendental number, consists of only $0$'s and $1$'s, so it is clearly not normal in base $10$. $\endgroup$ – KSmarts Jan 14 '15 at 22:28
  • $\begingroup$ If $\pi$ were to contain every possible sequence of numbers, it would surely contain the digits of $\pi$ (somewhere after the first trillion) and if this were true, $\pi$ would be to busy eating $\pi$es to be containing every sequence. Hence, a contraceptive :) $\endgroup$ – Nick Jan 14 '15 at 22:46
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Given that transcendental numbers (or, for that matter, irrational numbers) are infinite, non-repeating sequences of digits, they contain any possible sequence of numbers, including the one Arroway found.

This is not necessarily true. A number whose digits contain, with equal frequency, all possible sequences of the same length, is a normal number. There are trancendental numbers that are known to be normal, and others that are known to be not normal, generally because they have been specifically constructed as such. However, whether pi is normal has been long suspected, and appears to be the case when you look at the digits already computed, but it's never been proven one way or the other.

So, in that sense, it is indeed possible for there to be some kind of message encoded in pi that we would recognise as being something other than pure chance. It's also one of the best places to put such a message, since pi is (believed to be) a universal constant that does not depend on local physics or units of measurement. However, there is a risk that other sentient races would use a different circle constant (such as some movements on Earth that claim that tau, $\tau=2\pi$, is a more natural choice) and any race that goes down that route would never see such a message (and if the message is actually hidden in tau, we won't see it either).

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  • $\begingroup$ Unless the decimal representation of $\tau$ contains all sequences of digits that can be found in $\pi$, which is possible even with none of them being normal. $\endgroup$ – Henrik Jan 14 '15 at 22:27
  • $\begingroup$ As I think Henrik is alluding to, the relevant property you want is not normality, but rather the much weaker propery of simply having every finite digit string appear at least once (and hence, infinitely often) in the decimal expansion of ${\pi}.$ $\endgroup$ – Dave L. Renfro Jan 14 '15 at 22:40
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    $\begingroup$ By the way, for a couple of assertions that can be made about any irrational number, see my 2nd and 3rd applications of the pigeonhole principle in my answer to Examples and applications of the pigeonhole principle. $\endgroup$ – Dave L. Renfro Jan 14 '15 at 22:45
  • $\begingroup$ @DaveL.Renfro: I admit, it wasn't too clear what I meant. I was taking about the even weaker property of containing the same finite digit strings as another number. E.g. $0.\overline{12}$ and $0.\overline{21}$ contains the same finite digit strings. $\endgroup$ – Henrik Jan 15 '15 at 11:26
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Being transcendental doesn't mean every possible sequence of digits will appear.

That is the case for so called normal numbers. I believe $\pi$ is suspected to be normal in any base, but as far as I know that hasn't been proven yet.

If indeed $\pi$ is normal in base 11 (I have no idea whether that is known, according to wikipedia that is unknown) she would indeed be able to find a sequence of digits that looks right.

Note: I haven't read the book, I've seen the movie and didn't particularly like it, t were told by one of my friends (who have read the book) that the problems I saw in the movie (I don't really recall the details) were explined a lot better in the book.

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While you are correct that every possible pattern will eventually be found, it would take trillions and trillions of years to find the more complex patterns. If you read the book closely, you'll see that the "circle" pattern was found much, much earlier than this, and that's why it implies the "Artist's Signature".

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  • $\begingroup$ As said, we do not know whether we can find every pattern because we do not know whether every pattern is in $\pi$. But it is true that finding it would take almost an eternity in general. $\endgroup$ – Peter Jan 9 at 16:51
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I think the point is that it's would be a highly unusual finding based on the part of Pi we've seen until now. I don't know how big the pattern was but it was the product of two prime number indicating a structure - lets just say it was 121 numbers long (11x11).

That would mean there was a sequence of 121 number that was mostly 0 with some 1s. The question is, how often does such a thing happen - not how often could it happen with an infinite number of numbers - but how often has it happened in the 10^20 numbers they know in the movie.

If it happens all the time.. Most of the number in Pi are 0 or 1 and this time it happened to make a circle - you could say - it's just chance.

But if it's unusual to have a sequence that is only 0s and 1s and on top of that the first/only time we found it formed a picture of a circle - then you should realize this isn't an accident.

To claim that, "but there are still an infinite numbers to look at so probably this happens all the time" is not being intellectually honest.

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Let us assume that pi is a normal number (a reasonable, yet unproven conjecture). Then yes, any combination, specifically the one envisioned by Sagan, is buried somewhere deep inside pi. In this sense there is nothing surprising. However, the surprise would be to find it much sooner than expected. That is the main meaning of Sagan's idea: Not that Ellie found it, but that she found it so early that it would be statistically so improbable, that it would be nearly impossible to occur naturally.

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Actually, the aliens (call them $A$), were in the process of making $\pi$ rational starting at the Feymann point,

$$\pi = 3.1415926\dots9999998\dots$$

but they were rudely interrupted by the approach of the Death Star.

Seriously, regarding the whole question of a message being inserted in the digits of $\pi$ by an alien intelligence, this is covered by the post,

  1. "Could pi have a different value in a different universe?"

as the question is essentially,

  1. "Can pi have a different value at different times (pre-$A$ and post-$A$) in the same universe?

The excellent answer to the first also applies to the second.

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  • $\begingroup$ Why should we even assume that there is another universe ? $\endgroup$ – Peter Jan 9 at 17:02

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