A model problem, based on the NACA0012 airfoil, is studied. The thickness distribution of the symmetric NACA0012 airfoil is optimally approximated with a 4 th-order Bezier curve. This best-fit approximation is designated Bez4-0012. Introducing this approximation is similar to the first step taken in optimization methods which work on an absolute analytic definition of an existing geometry, rather than on a perturbation of the baseline shape. Advantages and disadvantages of these approaches are discussed. In order to systematically study the impact of dimensionality, an infinite family of design spaces is developed. This family has the property that all airfoil shapes supported in its M-space are fully and exactly contained within its N-space, where 3 ≤ M ≤ N. Response-surface and gradient-based methods of optimization are applied. Results for design-space dimensions of [3, 6, 12, 24, 36] are presented. These data include trends with respect to increasing dimension on realized minimum-drag designs, and computational costs.