  We report a novel and spectacular instability of a fluid surface in a
rotating system. In a flow driven by rotating the bottom plate of a partially
filled, stationary cylindrical container, the shape of the free surface can
spontaneously break the axial symmetry and assume the form of a polygon
rotating rigidly with a speed different from that of the plate. With water we
have observed polygons with up to 6 corners. It has been known for many years
that such flows are prone to symmetry breaking, but apparently the polygonal
surface shapes have never been observed. The creation of rotating internal
waves in a similar setup was observed for much lower rotation rates, where the
free surface remains essentially flat. We speculate that the instability is
caused by the strong azimuthal shear due to the stationary walls and that it is
triggered by minute wobbling of the rotating plate. The slight asymmetry
induces a tendency for mode-locking between the plate and the polygon, where
the polygon rotates by one corner for each complete rotation of the plate.
