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?
Best Answer
The symmetry group of the octagonal gluing is an order 16 dihedral group. But the symmetry group of the depicted embedding is an order 12 group which is the direct sum of an order 6 dihedral group with an order 2 cyclic group.
Given that the symmetry groups are not isomorphic, in fact not even of the same order, what you're asking for seems impossible to me.