I am interested in 3-D representations of various things that naturally live in a non-simply-connected compact surface. There is the usual way of producing a compact surface of any orientable or non-orientable genus, by associating a boundary word to the edges of a polygon to indicate how they are to be glued. But there may or may not be an isometric embedding of this polygon quotient into R^3, not even a local one (isometric immersion). As a simple example : "a a^-1 b b^-1 c c^-1" generates a simply connected surface, but if you put this word around a regular hexagon and fold accordingly, the result is a squashed tetrahedron - not even an immersion. You can make it work by altering the angles of the hexagon slightly.

Is there a known method to decide, given the angles at each vertex of the unfolded polygon, and a boundary word, whether an isometric embedding, or immersion, exists? (More ambitiously I would like it to decide whether it is ugly, but that sounds like it might be an AI-complete question.)

### Addendum

The faces are to be pasted as Joseph O'Rourke indicated. Thanks for the clarification.

So yes, there would be a finite number of points with angle other than 2 pi (they could actually be greater - multiple vertices of the original polygon can get identified.) Everywhere else it's flat. Self-intersection is unavoidable for non-orientable cases, but for example the regular hexagon case, the "embedding" is not injective even locally.

And never mind "ugly". A decision procedure for that would probably be a sentient algorithm (AI-complete question).

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