TITLE: Triangulation of the Klein Bottle
QUESTION [7 upvotes]: Why is this no triangulation of the Klein Bottle?

Is it because the top and the bottom triangle share 3 vertices but have different edges?
How do I find a triangulation?

REPLY [4 votes]: The vertices of the outer square do not remain distinct after identifying edges to create the Klein bottle.  So some of your "triangles" have less than $3$ vertices.