Problems

If we glue the opposite sides of the paper band in the same direction as on the picture, we will get a cylinder. What surface do we get, if we glue the circles of the cylinder in the same direction as well?

We start with a rectangular sheet of paper, preferably with proportions more than \(6:1\), so it looks more like a band, for now assume that one can stretch or shrink the paper band as needed. Describe the surface which we can get if we start with a rectangular sheet of paper and then glue the opposite sides of the paper band in the opposite direction as on the picture.

How would you describe the surface, obtained by glueing the sides of the octagon as on the picture? The sides of the same color are glued together in the same direction as shown.

Describe the surface which we can get if we start with a rectangular sheet of paper, make a cylinder by glueing the opposite sides in the same direction and then glue the other opposite sides of the paper band in the opposite direction as on the picture.