Variations on the Projective Central Limit Theorem

This expository article states and proves four, concrete, projective, central limit theorems. The results are known or suspected to be true by experts who are familiar with the more general central limit theorem for convex bodies, and related theory. Here we consider only four types of high dimensional geometric objects: spheres, balls, cubes, and boundaries of cubes. Each is capable of transforming uniform random variables into normal random variables through projection. This paper has been written to introduce new proof techniques, demonstrate how statistical simulation can be applied to geometry, and to build a foundation upon which recreational research projects can be built. The goal is to give the reader a better understanding of some of the mathematics at the juncture of probability theory, analysis, and geometry in high dimensions.