Intuitively, when we think of surfaces, we might think of ripples on the surface of a pond, a tabletop, or the outside covering of a globe. In a mathematical sense, most surfaces are like a plane bent smoothly in space (possibly with self-intersections, singularities, and other oddities.) That is, they are subsets of R³ that are in some sense 2-dimensional. More precisely, a surface is defined by a map from R² into R³.