Finding Waldo in π

> What are the odds?

Assuming pi is normal [1], the probability of any bit string occurring in pi is 1.

[1] https://en.wikipedia.org/wiki/Normal_number

All the 4 images posted by the author are located precisely at the begiging of a hexadecimal digit.

Since a hexadecimal digit is 4-bit long, I guess we should be able to locate 4x more Waldos if we allow solutions to be start in the middle of a hex digit?

> The pixel data in this gif are the 23,074,248th through 23,075,235th hexadecimal digits of π! (Equivalently, the 184,593,977th through 184,601,880th bits).

The author assumes a hex digit is 8-bit long here btw.

This post reminds me of the once common meme[1] that pi contains all things you will see in your life, all books you will ever read and all songs you will ever hear, the first thing you saw when you were born, and the last thing you will see before you go away for one final time.

Off course, pi isn't proven to be normal (containing all possible sequences of decimal digits, if I remember that correctly) yet so this thought experiment needs an encoding aware of that to work, and any random bit source that unrepeatingly explores the combinatorial space of a symbolic alphabet would work, pi is not special here at all. This is just the library of babel + encoding arbitary data structures into numbers.

As you say, finding patterns is not hard, it's unsurprising that random strings contain those things, maximum-entropy information sources maximize the expected information as per Shannon. The whole point, though, is finding useful patterns: things that predict other things you care about, which you didn't know beforehand (or just knew in vague outlines), and which are cheaper to find and explore than simply directly observing/simulating the things they predict. None of this holds for the 'patterns' you will find in a typical library of babel. Evolution and evolutionary algorithms can be seen as a tool to cull the impossibly large search space of a library of babel. Also mental heuristics like Occam's razor can be seen as tree-pruning heuristics this way.

An even more beautiful idea than "An infinite random string contains all possible data structures" is "An infinite random string contains all possible programs and all possible computations (according to all possible semantics of those programs)", explored by the legendary Greg Egan in Permutation City. I can't do justice to Egan in this already too-long-of-a-comment, but I promise you will absolutely be mind blown.

[1] https://slate.com/technology/2013/04/pi-meme-on-reddit-and-g...

That's pretty clever and impressive.

Could there be a number, such that for certain interesting data, you can compress the data more than normally achiveable, by indexing into the number?

I feel this should not be possible, but I don't know how to proove it without a circular argument about entropy.

On the other hand, if you make a string of common byte sequences, a couple GB long, and distribute it with every PC... then you surely can achive hyper compression with some clever indexing (if you don't count the magic string to the compressed size, of course).