  We observe experimentally the spontaneous formation of star-shaped surface
patterns in driven Bose-Einstein condensates. Two-dimensional star-shaped
patterns with $l$-fold symmetry, ranging from quadrupole ($l=2$) to heptagon
modes ($l=7$), are parametrically excited by modulating the scattering length
near the Feshbach resonance. An effective Mathieu equation and Floquet analysis
are utilized, relating the instability conditions to the dispersion of the
surface modes in a trapped superfluid. Identifying the resonant frequencies of
the patterns, we precisely measure the dispersion relation of the collective
excitations. The oscillation amplitude of the surface excitations increases
exponentially during the modulation. We find that only the $l=6$ mode is
unstable due to its emergent coupling with the dipole motion of the cloud. Our
experimental results are in excellent agreement with the mean-field framework.
Our work opens a new pathway for generating higher-lying collective excitations
with applications, such as the probing of exotic properties of quantum fluids
and providing a generation mechanism of quantum turbulence.
