# Minimal triangulation of Klein bottle

What is minimal triangulation of Klein bottle? А triangulation is a subdivision of a geometric object into simplices. Minimal in sense of vertex count.

So, I know that minimal count of vertex in the shortest triangulation must be greater then $7$, because the shortest triangulation of torus consist of $7$ vertex and Euler characteristic is equal to $0$.

I would be cool if you can show me the picture.

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I'd be impressed if there was a way to do it in less than $8$. – Thomas Andrews Jul 27 '13 at 12:11
The minimal number is 8. Section 4 of this paper has a proof of that. Section 5 of same paper derive all six distinct 8-vertex triangulations of the Klein bottle and has picture for them. – achille hui Jul 27 '13 at 13:02
Thanks, its very good paper. – Gleb Jul 27 '13 at 13:21

The Klein bottle can be seen as the square $I^2$ with the boundaries identified in a specific way. Thus some triangulations of the square induces a triangulation of the Klein bottle. In particular you have a triangulation with exactly two faces, three edges and one vertex induced by the triangulation of the square obtained by cutting along the diagonal.
Unfortunately, this partition isnt triangulation, because we have a edge, which incidence only one vertex. So, it isn`t simlices. – Gleb Jul 27 '13 at 12:35