tombert

I know it was meant as a bit of a goof, but i always thought that the "look up the file in pi!" idea was a clever way of doing compression[1], exploiting the fact that pi is normal and thus any given sequence of numbers will happen eventually, meaning that any given file that could possibly exist will show up eventually. Using this, you could theoretically just store two numbers to compress anything, a beginning point and an end point for its position in pi.

I'm aware this would be unbearably slow and the numbers to store really big files might end up taking more memory than LZMA, so I don't need a lecture about it, I just thought it was an interesting idea.

[1] https://github.com/philipl/pifs

anuragbiyani
It's a commonly held misconception that Pi is proven to be normal - it is NOT (it is a conjecture as of now) [1] [2]. Proving normality of number is a very hard problem, and hardly any numbers outside of purposefully constructed ones (such as Champernowne's constant [3]) are proven normal.

In fact it's not even proven that every digit occurs infinitely many times in the decimal expansion of Pi. [4]

So https://github.com/philipl/pifs is wrong in claiming that all byte stream will exist somewhere in Pi (it's not proven). Also it's worth calling out that even if Pi was Normal it will likely take more space to store the indices of two location as it will for original data itself (for at least majority of the integers), so it's not much of a "compression" strictly speaking. It's easy to see how this will work out for a known normal number - Champernowne's constant [3] -> Unlike Pi, Champernowne's constant is guaranteed to contain all the possible natural number sequences, but storing just the starting index of them in this constant is going to take much longer than the entire number itself (e.g., number "11" start at index 12 (1-indexing), number "12" starts at index 14, and so on - the size of index increases much faster than integer being looked up itself).

[1] https://mathworld.wolfram.com/NormalNumber.html

[2] https://math.stackexchange.com/a/216578

[3] Champernowne's constant (in base 10) is the concatenation of all positive integers and treating them as the decimal expansion (following "0."): 0.12345678910111213... It can be trivially seen that it contains all natural number strings. It is also proven to be Normal in base 10 (which is a stronger property). See https://en.wikipedia.org/wiki/Champernowne_constant for details.

[4] https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...