  To explore basin geometry in high-dimensional dynamical systems, we consider
a ring of identical Kuramoto oscillators. Many attractors coexist in this
system; each is a twisted periodic orbit characterized by a winding number $q$,
with basin size proportional to $e^{-kq^2}.$ We uncover the geometry behind
this size distribution and find the basins are octopus-like, with nearly all
their volume in the tentacles, not the head of the octopus (the ball-like
region close to the attractor). We present a simple geometrical reason why
basins with tentacles should be common in high-dimensional systems.
