Abstract: For the last 5 years the focus of the Rose-Hulman REU Tilings group has been hyperbolic, kaleidoscopic tilings of Riemann surfaces by triangles. A lot has been discovered about these objects including a complete classification up to genus 13. Last summer we pushed beyond triangles to consider quadrilateral tilings. On the plus side the group theory did get a bit simpler; on the minus side we lost rigidity. A surface constructed from triangles is rigid in the sense that there are no transformations that preserve both angles and area. This is not true in the quadrilateral case. The euclidean analog is that all triangles with congruent corresponding angles and the same area are congruent. However, there is a one-parameter family of mutually non-congruent rectangles with the same area. On hyperbolic surfaces the same holds true, but there is an interesting twist. As we vary the quadrilaterals through an infinite family of equiangular, equal area quadrilaterals some curves on the surface take on arbitrarily small lengths, and shrink to a point as we go to infinity. These are the so-called "vanishing cycles" studied in algebraic geometry. We will show how to identify the vanishing cycles in simple geometric terms. Much of the talk will be explaining the basic concepts in terms of small visual examples. Students Isabel Averill, Michael Burr, John Gregoire and Kathryn Zuhr all contributed to this project.