Their surface area is given as 4 pi r squared.
Brouwer proved that every smooth vector field on one of these objects has a singular point. The Banach-Tarski paradox concerns the decomposition of one of these objects and reassembly into two identical copies. The complex plane plus a point at infinity is represented on one of these objects named for (*) Riemann. The azimuthal angle, polar angle, and radial distance are the three components to their namesake coordinates. Their surface area is given as 4 pi r squared. For 10 points, name these three dimensional analogues of circles.
Brouwer proved that every smooth vector field on one of these objects has a singular point.
spheres