As a mathematical layman, I would like to ask about a phenomenon that has intrigued me but maybe is quite trivial:
Consider the unit circle arc $y = (1 - x^2)^\frac1 2$ in the interval $0 < x < 1/\sqrt 2 $. If you take a line perpendicular to the tangent (here a radius) at any point on that arc, there will be a local mirror symmetry of the arc across that line. See image (n = 2).
Now let's look at other cases of $y = (1 - x^n)^\frac 1 n$ (n integer > 0) in the same interval (or in the intervals 0 < x < 1/(nth root of 2)). You will get a symmetry of the corresponding kind for n = 1 (yields a -45° line) and also when n goes to infinity (yields a horizontal line). n = 1, n = 2 and n $\rightarrow$ infinity are also the only cases where the graph will cross rational points. For all other n's, the resulting arcs will lack the symmetry described above, and (by Fermat's last theorem) they will never cross rational points.
To a mathematician, is there an obvious connection between these facts and FLT (or to the existing proof of FLT)? Or is it an equally obvious coincidence, or even a trivial thing? It's such a basic observation that I guess it has been investigated at some point in time – or has it?