Example 1: Footballs We'll turn the idea of slicing up a basketball to form a football into mathematical language. A rotation by 2Pi/8, fixing the north and south poles, moves a longitude by the angle 2Pi/8 to a new longitude. Look at the black wedge closest to the ring in the above picture of the Black and White Masterball. Any point outside this wedge can be rotated into it. This wedge is called a fundamental domain. The boundary longitude in between the black and white wedge closest to the ring gets rotated to the opposite longitude across the black region, so when we roll or sew up the basketball, these will be the same in the new space - a thin football. Visualize this as an orange peel wedge with the longitude edges sewn together (you can try this!), or by rolling an orange peel around itself eight times. The football has diameter Pi, the distance on the football between the sharp end points. The football is thin, but just as long as the sphere.
Example 2: Impossiball Another example is related to this Impossiball, a Rubik's Cube game. Imagine yourself sitting inside an icosahedron, a platonic solid with five triangles around every vertex point, which is made out of rubber bands and is sitting inside a basketball! Push the rubber bands until you can glue them to the basketball. The result is the Impossiball. Rotation by 2Pi/5 moves these triangles around. Playing with the Impossiball yourself helps with this visualization, so ask me to show it to you sometime. The symmetries of the icosahedron act on the basketball. Think about what a fundamental domain would look like. The sewed up space has diameter approximately Pi/4.8. It is the smallest space you can get by using rotations to slice up a basketball!
Definitions and Additional Reading for Undergraduates