# What surface forms when you glue opposite sides of a rectangle together, introducing a half twist for both gluings?

I'm aware that when you glue the ends of a rectangle together normally, you get a torus. I'm also aware that when you glue the ends of a rectangle together while introducing a half-twist to one of the sides, you get a Klein Bottle. The way that each of these things gets glued makes sense to me, and the homeomorphism is pretty clear.

But what happens when you try and introduce a half twist to both sides? Is it even possible to do this? I've been fiddling around with cut-and-paste methods to work out what this might be and I'm coming out inconclusive.

• The real projective plane? Aug 16 '18 at 15:42
• Aw, it is? That's so boring. Thanks, that ought to be enough for me to look more into it. Aug 16 '18 at 15:44
• Why is that boring? Aug 16 '18 at 15:45
• Note that unlike the torus and Klein bottle cases, all four corners of the rectangle do not come together by this gluing. Aug 16 '18 at 15:47
• Oh, wait a minute, the real projective plane might be something totally different from what I thought it was. At first glance, I thought it was just $\mathbb{R}^2$ projected onto a surface. Aug 16 '18 at 15:49