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!"?