In the hunt for an optimal geometric object, one way to proceed is to start with a non-optimal object and modify that object continuously in an "optimizing direction." As an example: a natural way to consider the Isoperimetric Problem (the problem of finding, among all closed curves in the plane with the same perimeter, the curve that encloses the largest area) might be to start with any closed curve, and then continuously evolve this curve to enclose more area.

Using the Isoperimetric Problem as a starting point, we explore the process of using variations to identify optimal geometric objects. We introduce Minimal Surfaces and Constant-Mean-Curvature Surfaces as classical examples, before turning towards recent developments in finding isoperimetric regions in probability (Gaussian) space.