# How to construct a Klein Bottle

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.

## The Torus

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.

### Construction

1. You will need the following items:
• A sheet of paper; (Tissue paper may be easiest)
• Sticky tape or other adhesive;
• Doughnut mix. (Optional)
2. Tape one pair of opposite sides together so that you have a cylinder.
3. Curl the other pair of opposite sides of the paper and stick them together. You should now have a torus.
4. Fill your torus with doughnut mix and fry to recreate a doughnut. Do not eat it.

### Analysis

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.

## The Klein Bottle

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.

### Construction

Basic instructions for the Klein bottle. Large monetary rewards for anyone who succeeds and is willing to sell.

1. You will need the following items:
• A sheet of paper;