Is it possible at every point $p=(x,y)$ on the unit circle, there is a continuous curve $C_p$ passing through it, a curve which is not only the single point $p$, and all these curves are pairwise disjoint?
And there is a constant $e$, such that all these curves have length longer then $e$.
Edit:
How to prove that they must be all the same curve locally around $p$, except translated and rotated?