GeometryThe chief geometric question we explore is: What happens to a path or geodesic that begins at a corner? Does it reach another corner? If so, which? We will organize our work by establishing a standard position for looking at paths from a corner. We'll also keep track of all possible box shapes using something we call the box triangle. We'll use color to code the corners (the possible destinations) of the paths. Sometimes we'll describe the path as the result of rolling the box across a plane. Also useful is a utility? that allows you to choose a box, look at the path from the corner, and rotate the box to see it from all sides. Boxes are an example of translation surfaces which have been studied more generally. Boxes with integral dimensions are examples of square-tiled surfaces. Our explorations are on these pages: |