Okay I have a math question.

So, take a diagonal path on a grid (say a grid of streets).

If you're constrained to the grid, you'll go the same length no matter which way you take, right?

Because all those ups and overs can just be rearranged outward into a single "up, over, done".

And you can keep doing this with smaller and smaller squares. The path length doesn't change, you're just rearranging shit.

But then why is the 90° path longer than the diagonal?? At infinity, it would suddenly JUMP to being shorter, which feels a bit odd. I'm not sure how to reconcile this.

@frostwolf the "jump" to the diagonal doesn't make sense here, you're trying to approximate the diagonal but you're never reaching it

@noiob Ohhh yeah! That makes sense.

*facepaw* of COURSE the ratio between a square's sides and diagonal doesn't change with how big the square is. And while you're shrinking them, you're also adding more squares, so it never goes anywhere.

Even though it LOOKS like it should, since it more closely approximates the path.

@noiob so like at infinity, you have infinitely small squares, but you also have infinitely many of them!

I guess this is just one of those cases where the limit doesn't actually equal the value at infinity/0.

@frostwolf kinda, but infinity isn't a number, mathematically "at infinity" doesn't make sense