We have already see this: the space of rotations of the sphere. As well known in 3D graphics, this space is the same as unit quaternions, i.e., a 3-sphere (with opposite points identified). Today, we will modify this to get a new geometry! (1/6+)
The straight lines you see are called fibers. We change the geometry by stretching the metric along the fibers. As if the speed of light was slower/faster along the fiber, and light still took the fastest route, as usual.
Here we stretch the fibers to just 2%. There is only one white block on every fiber -- but since it is so fast to travel along the fiber, we see many copies of every white block! In the limit we get the S2xE geometry.
We missed one cool feature of this geometry! Without stretch, if light ray nearly misses a block, it travels π units around the sphere, also nearly misses it again. With stretch, it may hit another block. Here we show only white blocks, not the whole fibers.
Here we stretch the fibers to 10x. The afterimage effect seems to be caused by errors in numerical solving of differential equations (it gets smaller when precision is increased), but it is still cool.
The small image in the corner shows the rotation corresponding to the current camera position. Thanks to Jeff Weeks for suggesting this geometry, and some hints! YouTube link:
