You’re probably thinking “I know what a sphere is; it’s the locus of all points a fixed distance from a center.” The definition you're thinking of is accurate, but you might want some other ways to distinguish spheres from other surfaces. I will discuss some geometric properties of surfaces which seem like good characterizations of spheres. Some of them are, and some are not.

To understand this lecture, you will need to know the Pythagorean theorem (to compute distances), and possibly how to differentiate and integrate. It might also help to draw some pictures of spheres and other surfaces. (Go on, try it now!)