If you're a mildly curious person without a deep understanding of topology you might be thinking to yourself "what is that very cool bottle shaped object and how can I get one?" It's called a Klein bottle and it's the simplest example of a closed, non-orientable manifold but we'll get to that later. A Klein bottle can be tricky so we'll start with something simpler, the doughnut.
To understand what we do to make a torus you first should understand how to undo it. Imagine a doughnut. Take a knife, slice all the way around it (not through to the other side) and scrape out the filling. If you fold the "skin" back to where it was before you get a torus - the outside of the doughnut. Eat the filling if using a real doughnut. You now have a thin cylinder of doughnut skin. Cut it and lay it out flat. This is our sheet of paper.
Follow these steps to get your very own torus.
Notice that when you join a pair of opposite sides it ceases to be a border. When you join up all the sides you can draw a line between any two points and the edges of the paper stop being an obstacle. However, you can still choose a side of the paper. That means that you can choose one point on each side of the paper and draw as long a line as you please, you'll never cross over.
And now what you've all been waiting for, our friend the Klein bottle. The Klein bottle is like the torus but without the inside.
Basic instructions for the Klein bottle. Large monetary rewards for anyone who succeeds and is willing to sell.
Did you actually try it? If you did, good for you. If you succeeded, contact me for fame and fortune. The reason most of us do not succeed is quite simple. I'll illustrate it with an example. Take a piece of string. Make a circle with it. Wasn't that easy? Now try to make a figure 8. This is what you did with the Möbius strip. Unfortunately, to make a Klein bottle there is a catch. You have to be able to make a figure 8 with a single piece of string that doesn't cross over itself. There is no up or down - only across. That's what happens when you restrict yourself to two dimensions.
The Klein bottle is much more interesting than the torus because you can draw a line between any two points on either side of the paper. You can see this because you can do it with the Möbius strip too. The extra joy in the Klein bottle comes from the fact that is has no edges. Just like the torus, it's all smooth. The cylinder has two circular edges for paper cuts. If you're wondering, closed means no edges to fall off. Non-orientable means that if you move an arrow around the Klein bottle, you can get back to where you started with the arrow pointing the wrong way (even though it's on the other side of the paper). Manifold just means surface.