In the Euclidean plane, is the circle the only simple closed curve that has an axis of symmetry in every 
direction?



Can the circle be characterized by the following property?
A slightly different argument is as follows. Choose two symmetries $\sigma,\tau$
with axes
intersecting at a point $P$ and forming an angle of $2\pi \lambda$ with $\lambda$ irrational.
The composition $\rho=\sigma\circ \tau$ is then a rotation of infinite order generating
a dense subgroup of the group of all rotations centered at $P$.
Any closed subset left invariant under $\rho$ is thus a union of concentric circles centered at $P$. A simple closed curve invariant under $\rho$ is thus such a circle.


