system
Given the following post, respond as if you're commenting on Reddit:
human
An infinitely long line made into a circle?

My question might be odd but i can't figure it out or come to a conclusion. 

So if i had an infinitely long line and turned it into a circle, the circle is infinitely long but has to end because a circle is a closed loop. Infinity can't end tho, as it goes on forever? 

Or am i just misunderstanding something?

Any thoughts and answers are appreciated, thanks!
gpt
In math, there are many different notions for "geometry" and how to talk about shapes. What we're generally used to from high school math is "Metric Geometry", which has to do with lengths and distances and such. On the other hand, there is a type of geometry known as "Topology" which does not give a shit about lengths and distances and only cares about "shape".

If you have an object in Metric Geometry, then you can move it around and rotate it and it won't change. These are "isometric" transformations because they don't change lengths or anything, and so Metric Geometry says that they're actually identical. If you have an object in Topology, you can also move it around and rotate it, but you can also stretch it, bend it, compress it, flip it in a mirror, etc you can do almost anything as long as you don't tear it. So Topology sees more things as being the same than Metric Geometry does. Eg, a square and a circle are the same thing Topologically, but not the same thing "Isometrically" (ie in Metric Geometry).

Now, to the question at hand. Say that you have the interval (-pi,pi). It's short, so you can first cap it off into the closed interval [-pi,pi] and loop it into a circle by gluing together the end points. You then get a normal circle of radius 1. This isn't done, specifically, through rotations and translations, but it does preserve lengths in our object. Now, here's the thing, since length doesn't matter to topology, the intervals (-pi,pi) and (-∞,∞) are the same topologically. In fact, the function 2arctan(x) compresses (-∞,∞) onto (-pi,pi) perfectly topologically. So, topologically, you can totally redo this process of creating a circle beginning with (-∞,∞) because, to topology, it's not infinitely long, it doesn't even make sense to talk about length, and it's exactly the same thing as (-pi,pi).

So, through the process you describe, you can create a circle when you look at things through topology, and that's good. It's also a square too. But if we track the lengths of things, then we don't get a circle through the lens of Metric Geometry. This is because it doesn't have a well-defined radius or circumference that is constant throughout. Moreover, we can try to associate angles with arclengths, while keeping length same throughout, and we get a nonlinear relationship between angle and arclength, going to infinity at the "top" point of the circle, which is not a property that circles have in metric geometry.

So, yes, you can make a circle. But you have to be careful about what context your circle exists in.