  We construct an approximate solution for an asymptotically flat, neutral,
thin rotating black ring in any dimension D>=5 by matching the near-horizon
solution for a bent boosted black string, to a linearized gravity solution away
from the horizon. The rotating black ring solution has a regular horizon of
topology S^1 x S^{D-3} and incorporates the balancing condition of the ring as
a zero-tension condition. For D=5 our method reproduces the thin ring limit of
the exact black ring solution. For D>=6 we show that the black ring has a
higher entropy than the Myers-Perry black hole in the ultra-spinning regime. By
exploiting the correspondence between ultra-spinning black holes and black
membranes on a two-torus, we take steps towards qualitatively completing the
phase diagram of rotating blackfolds with a single angular momentum. We are led
to propose a connection between MP black holes and black rings, and between MP
black holes and black Saturns, through merger transitions involving two kinds
of `pinched' black holes. More generally, the analogy suggests an infinite
number of pinched black holes of spherical topology leading to a complicated
pattern of connections and mergers between phases.
