Given a metric space and a positive connectivity parameter, the Vietoris-Rips simplicial complex has a vertex for each point in the metric space, and contains a set of vertices as a simplex if its diameter is less than the connectivity parameter. A theorem of Jean-Claude Hausmann states that if the metric space is a Riemannian manifold and the connectivity parameter is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. What happens for larger connectivity parameters? We show that as the connectivity parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy type of the circle, the 3-sphere, the 5-sphere, the 7- sphere, ..., until finally it is contractible. Joint work with Michal Adamaszek, Florian Frick, Christopher Peterson, and Corrine Previte.