Polygonal decomposition of the klein bottle by pentagons?

Question

How do I prove that there is no regular polygonal decomposition of the klein bottle by pentagons?

Do I use the euler characteristic to prove?

Answer 1

I don´t know what "regular" means when talking about polygonal decompositions, but I´ll write an answer ignoring that word in case it helps. If it´s wrong I´ll delete it.

I don´t see why you couldn´t do what you ask. In the figure you can see a decomposition of a triangle into 4 pentagons. This construction uses as vertices of the pentagons the middle points of the edges of the triangle, but the construction can be easily modified so that you use whatever points you want (1 of each side of the triangle) as vertices of the pentagons. So it seems any triangulation of the klein bottle can be converted into a polygon decomposition in pentagons. You just have to take care to, when you have two triangles adjacent in an edge, use the same point of the edge to divide the triangles into pentagons.