  We find a very large set of smooth horizonless geometries that have the same
charges and angular momenta as the five-dimensional, maximally-spinning,
three-charge, BPS black hole (J^2 = Q^3). Our solutions are constructed using a
four-dimensional Gibbons-Hawking base space that has a very large number of
two-cycles. The entropy of our solutions is proportional to Q^(1/2). In the
same class of solutions we also find microstates corresponding to zero-entropy
black rings, and these are related to the microstates of the black hole by
continuous deformations.
