Theoretical pi-based stream cipher Let's pretend that all digits of pi are known and arbitrarily long sequences of digits are trivial to get. Further, some mathematician proves that there are no patterns in pi. We could create a stream cipher by grabbing a piece of pi as long as our plaintext and combining the two with some function (such as XOR or modulus addition.) The key would be the starting position in pi.

Would this be equivalent (in terms of security) to a one-time pad? To what sort of attacks would it be vulnerable?
The problem with this approach is that it literally gains you nothing. In order to choose a random subsequence of a needed length from $\pi$, you need to generate a cryptographically random number of at least the same length of the desired key to use as the offset. But then you may as well just use that number as your secret key.

Other than that, yes, it's exactly the same as a one-time pad. Just with a silly and pointless key derivation protocol which cannot mathematically increase the security of the system, but could conceivably weaken it.

Edit: As Thomas points out in the comments, the distribution of digits of $\pi$ are not random, and so this mechanism of key generation actually discards a significant amount of entropy that had been generated while choosing a random offset.