The genetics and evolution of biological systems are extremely complex because of the large number of traits , and complex relationships among those traits. We use the form of fruit fly wings as a model to study the variational properties of complex biological structures. Variation is important because it controls evolutionary potential. Questions about evolutionary potential of high-dimensional entities raise a series of difficult mathematical and statistical problems. Our data suggests that the dimensionality of the underlying system is very high. Could the data lie on a manifold embedded in the linear space of phenotypes? If so, phenomena that seem complex could have simple explanations. Manifold-finding based on genotypic data has not yet been attempted. The pattern of variation in two different populations can be quite different. Can we identify the common phenotypic subspace, and, even more interesting, the subspaces where one has variation, and the other does not? Statistical approaches to those questions are not known (at least in biology) How can we understand and predict the appearance of qualitatively novel phentoypes? Qualtitative novelty is one of the largest unsolved problems in biology. Is it possible to construct metrics for the ?novelty distance? between phenotypes that predict evolution? One possible kind of metric could combine geometry and topology as is done with persistent homology. Biology may offer different metrics based on the effects of mutation or common transitions during development. Biologists need the expertise of mathematicians and statisticians to help us answer these important questions.