Real Curves/Circles Unfortunately I am very ignorant when it comes to mathematics. Please understand and forgive me if this question reflects that. Thank you!
I have an observation: Every curve and every circle in the real world is really just a bunch of straight lines and angles. There is no “real” curve that can be seen.
Take any circle. At some point in the circle, however miniscule, is a straight line. Even if to see it you would have to use a microscope that is so powerful it doesn’t exist, in theory the straight line has to be there. If it weren’t, the curvature of the circle would be so sharp that it would converge on itself. Now take it one step further. If you’d take a microscope to any point in a curve, if you take a small enough space, you’ll find a straight line. For the same reason – if it isn’t straight even as you go infinitely small, then at that point it has to keep curving infinitely, which means it should converge on itself. Since it doesn’t, there must be a straight line there. To get to the next point, also a straight line, there must be an angle.
Is this correct?
 A: No, that is wrong all the way.
Most of the curves we consider in mathematics do not consist of straight line segments meeting at angles. In particular a circle does not consist of straight line segments meeting at angles -- it really is curved everywhere.
A circle is by definition the set of all points that have a certain fixed distance to a fixed point, the center of the circle. If, anywhere on the circle, it contained a straight line segment, that would mean that both the endpoints and the midpoint of that line segment would have the same distance to the center. But it is impossible for three different points on the same line to have identical distances to a fixed point (this can be proved in various ways, depending on which axioms you accept). Therefore there is no line segment that is part of a circle.
You argument:

If it weren’t, the curvature of the circle would be so sharp that it would converge on itself. 

does not make sense without a definition of what you mean by "converge on itself". An even so, it would need some kind of argument, rather than a bare assertion that any non-polygonal curve will "converge on itself".
