That’s not how it’s works. Being “infinite” is not enough, the number 1.110100100010000… is “infinite”, without repeating patterns and dosen’t have other digits that 1 or 0.
If it’s infinite without repeating patterns then it just contain all patterns, no? Eh i guess that’s not how that works, is it? Half of all patterns is still infinity.
However, as the name implies, this is nothing special about pi. Almost all numbers have this property. If anything, it’s the integers that we should be finding weird, like you mean to tell me that every single digit after the decimal point is a zero? No matter how far you go, just zeroes forever?
In some encoding scheme, those digits can represent something other than binary digits. If we consider your string of digits to truly be infinite, some substring somewhere will be meaningful.
One of the many things I loved about Sagan’s Contact is that, at the end, they found a pattern in pi when put into base 13. He didn’t really go into it as it was the end of the book, but I really wish he’d survived to write a sequel.
to be fair, though, 1 and 0 are just binary representations of values, same as decimal and hexadecimal. within your example, we’d absolutely find the entire works of shakespeare encoded in ascii, unicode, and lcd pixel format with each letter arranged in 3x5 grids.
Actually, there’d only be single pixels past digit 225 in the last example, if I understand you correctly.
If we can choose encoding, we can “cheat” by effectively embedding whatever we want to find in the encoding. The existence of every substring in a one of a set of ordinary encodings might not even be a weaker property than a fixed encoding, though, because infinities can be like that.
Still not enough, or at least pi is not known to have this property. You need the number to be “normal” (or a slightly weaker property) which turns out to be hard to prove about most numbers.
> natural numbers
> rational numbers
> real numbers
> regular numbers
> normal numbers
> simply normal numbers
> absolutely normal numbers
Have mathematicians considered talking about what numbers they find okay, rather than everyone just picking their favorite and saying that one’s the ordinary one?