The similarity of a given manifold to one of these objects can be measured with the Willmore energy. A vanishing at rank 4 (-5[1])higher than their dimension is among few established patterns in these objects’ chaotic higher homotopy (15[1])groups. A 3-dimensional one of these objects (15[1])is expressed (15[1])as a fiber (15[1])bundle (15[1])over (15[1])a 2-dimensional one (15[1])in the Hopf (15[1])fibration. (15[5]-5[1])John (15[1])Milnor constructed an “exotic” class of these objects, whose (15[2])eversion was described by Stephen (15[2])Smale. (15[2])Any (15[1])simply-connected closed 3-manifold is homeomorphic to one (15[2])of these (15[2])objects (15[7])by (*) Perelman’s (10[1])proof of the Poincaré conjecture. (10[2])A coordinate system (10[1])named for these objects (10[4])uses azimuthal (10[3])and polar (10[2])angles (10[1]-5[1])and has a Jacobian from (10[2])Cartesian coordinates of rho squared (10[1])times sine (10[1])of phi. (10[1])For 10 points, name these shapes consisting of all points a fixed distance (10[1])away from (10[1])their center. ■END■ (10[5])