Patterns in pi in "Contact" 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!"?
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).
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,


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


as the question is essentially, 


"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.