The original theory of splines grew out of the study of simple variational problems. A spline was a function that minimized some notion of energy subject to a set of interpolation constraints. A more recent method for creating splines is subdivision. In this framework, a spline is the limit of a sequence of functions, each related by some simple averaging rule. This paper shows that the two ideas are intrinsically related. Specifically, the solution space to a wide range of variational problems can be captured as spline spaces defined through subdivision.