Those are really big numbers, but while they start out large, the functions /x\, [x], …, still grow more slowly than A(x,x), and are pitifully small compared to BB(x). I’m not actually sure where S(x) = {x-sided shape of x} fits in, except that it’s computable, so still smaller than BB(x) (for all sufficiently large values of x).
EDIT: I misunderstood the definition of (x) as a circle, and thus as the end of a limiting process. The op intended it to be a polygon with [X] sides. The next paragraph is not valid, although the final one is.
However, (x) is not a number, it’s a limit, and more importantly it is infinite for x > 1. Since you were in an analysis class, this should have been talked about!
What I’m really confused by is
Imagine drawing a line from that point to one on the near surface of the sun, which represents infinity. (2) lies within the parentheses surrounding that period, and that’s an understatement of how close it is to zero.
Not only is (2) an infinite quantity, if it wasn’t it could be wherever you wanted to put it on the line. If you’re mapping 0 to infinity onto a finite length line, you can approach the limit any way you desire. Why not put 1 at the halfway mark and 2 at the three-quarter mark? That seems to me to be conceptually most simple. It’s not like we can have a 1-1 mapping of numerical increase to distance!
It is; I misunderstood, although I don’t think your notation is blameless.
Basically, in the sequence triangle->square->pentagon->… appears to be a process that approaches circle as the limit of the number of sides tends towards infinity. My first (and second) time reading through the article I missed that (x) is not circle of x, but rather the [x]-gon of x.
Those are really big numbers, but while they start out large, the functions /x\, [x], …, still grow more slowly than A(x,x), and are pitifully small compared to BB(x). I’m not actually sure where S(x) = {x-sided shape of x} fits in, except that it’s computable, so still smaller than BB(x) (for all sufficiently large values of x).
EDIT: I misunderstood the definition of (x) as a circle, and thus as the end of a limiting process. The op intended it to be a polygon with [X] sides. The next paragraph is not valid, although the final one is.
However, (x) is not a number, it’s a limit, and more importantly it is infinite for x > 1. Since you were in an analysis class, this should have been talked about!
What I’m really confused by is
Not only is (2) an infinite quantity, if it wasn’t it could be wherever you wanted to put it on the line. If you’re mapping 0 to infinity onto a finite length line, you can approach the limit any way you desire. Why not put 1 at the halfway mark and 2 at the three-quarter mark? That seems to me to be conceptually most simple. It’s not like we can have a 1-1 mapping of numerical increase to distance!
(2) is clearly a finite quantity. I’m not seeing how you can think otherwise unless I’ve seriously miscommunicated something.
It is; I misunderstood, although I don’t think your notation is blameless.
Basically, in the sequence triangle->square->pentagon->… appears to be a process that approaches circle as the limit of the number of sides tends towards infinity. My first (and second) time reading through the article I missed that (x) is not circle of x, but rather the [x]-gon of x.