## Rubik's Cube Games on Spheres

Diameters of Quotients of
Spheres

Sarah
J. Greenwald
**Diameters**
In my research, I slice up basketballs (higher dimensional ones too!)
in order to form new spaces like a football, and then I measure
the diameters of the resulting spaces.
The diameter measures how far apart two electrons or fighting
ants can get away from each other on a space. Having a small diameter
is interesting, because it means that you can not escape far away
from an angry ant on the new space.
On a basketball of radius one, the diameter is Pi, the distance
from the north to the south pole, since we must travel on
the basketball instead of drilling through it.
Let's look at some examples:
**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!