This video is a representation of the octagon-to-genus-2-surface process. But there's a more symmetrical representation seen here:
The surface we see in the above picture isn't as symmetrical as it could be though (and the way it is glued up isn't as simple and symmetrical as in the video above either).
I want to make a nice video + animation about genus 2 surfaces (and their connection to the hyperbolic plane). The maximal symmetry on a genus 2 surface in (in euclidean 3 space) is like one of these:
Can anyone think of a simple, ideally highly symmetrical way of constructing either of those from the octagon at the top there?