I felt like adding something about the specific case of 180° between edges and a vertice.
Makes sense.
And I guess too many vertices means an open set of edges (ie not close, this not a shape).
I was kinda hoping for a strange edge case, like a mobius strip or Klein bottle.

I guess a mobius strip is a 2d representation of a 1d paradigm. And a klein bottle is a 3d representation of a 2d paradigm.
It would be too much to ask of a 1d representation of a ??d paradigm.