In the nineteenth and twentieth (and twenty-first!) centuries, mathematicians have been stretching the idea of “geometry” far beyond the geometry of Euclid most people are familiar with: into the fourth (or higher) dimension, curved spaces, and more. This new geometry (the part I am referring to is called “differential geometry and topology”) is philosophically and aesthetically interesting, plays a definite role in the construction of our universe, and has wide-ranging applications; but it is not well-known outside of mathematics departments. Usually, the prerequisites for this study are at least linear algebra, multivariable calculus, and analysis, so math majors get to it in their final undergraduate year, if at all. In this class, we will study these ideas in spaces made out of flat pieces (for a simple example, the surface of a cube). This will allow us to study sophisticated ideas, without assuming any background knowledge. In particular, I will not be assuming that students know any calculus; as for Euclidean geometry, we will be revisiting it from this larger perspective, so you do not need to know or remember that subject either. The class is open to everyone, but culminates in serious, high-level mathematics.