The similarity of a given manifold to one of these objects can be measured with the Willmore energy. A vanishing at rank 4 higher than their dimension is among few established patterns in these objects’ chaotic higher homotopy groups. A 3-dimensional one of these objects is expressed as a fiber bundle over a 2-dimensional one in the Hopf fibration. John Milnor constructed an “exotic” class of these objects, whose eversion was described by Stephen Smale. Any simply-connected closed 3-manifold is homeomorphic to one of these objects by (*) Perelman’s proof of the Poincaré conjecture. A coordinate system (10[1])named for these objects uses azimuthal (10[1])and polar angles and has a Jacobian from (10[1])Cartesian coordinates of rho squared times sine (10[1])of phi. For 10 points, name these shapes consisting of all points a fixed distance away from their center. ■END■