[pontifier]

The thing that blows my mind is: say you start filling the plane with pi. Pi has been proven to contain every finite sequence. That means that somewhere in the plane is a full physics simulation of YOU in the room you are in right now.

Does that you exist any less fully because its not currently in the memory of a computer being evaluated?

[7373737373]

> Pi has been proven to contain every finite sequence

This has not been proven yet: https://math.stackexchange.com/a/216348/575868

(or more generally: https://en.wikipedia.org/wiki/Disjunctive_sequence)

Depending on the infinite grid filling scheme even these properties may not be sufficient to guarantee that every two dimensional pattern is initially generated because the grid is two-dimensional, but the number property is "one-dimensional". A spiral pattern for example may always make it line up in a way such that certain 2d patterns are never generated.

[pontifier]

Since it's not provable with pi, then we'd have to do a more circuitous proof of every finite pattern occurring. Inspired by Champernowne's constant, I propose a Pontifier Pattern that is simple, inefficient, but provably contains every finite pattern.

Starting at the origin, mark off rows of squares. the Nth row would contain NxN^2 squares of size n x n. Each square would be filled in left to right reading order with successive binary numbers with the most significant digit at the top left.

Somewhere in that pattern is the physics simulation of you reading this comment :)

[7373737373]

minor correction: 2^(NxN) squares per row, right?

[pontifier]

Yeah, what was I thinking?? I really need to slow down sometimes. This should contain every finite pattern, right?

[7373737373]

Yes, sounds like it! Though I'm thinking that the relative arrangement of patterns would also make a difference. I wonder if such a thing as "all (infinitely many) possible arrangements of all patterns" can exist

[fsflover]

https://news.ycombinator.com/item?id=36357466