Ianagol on Nostr: When the cone angles on the spherical polyhedron are rational, there is a connection ...

When the cone angles on the spherical polyhedron are rational, there is a connection with translation surfaces. One can take a branched cover over the vertices so that the cone points all have cone angles a multiple of 2π, and so that there is a global holomorphic 1-form, hence a translation surface. Properties of this translation surface, such as closed geodesics or geodesics connecting vertices, can correspond to those of the polyhedron. For example, it underlies these results: https://arxiv.org/abs/1802.00811

https://arxiv.org/abs/1811.04131

https://mathemalchemy.org/2021/01/19/dodecahedral-trajectory/