Building Interesting Shapes by Gluing

I thought it would be fun to do a couple of strange shapes to show you the interesting things that you can do with a a bit of glue in topology. There are a couple of standard *strange* manifolds, and I'm going to walk through some simple gluing constructions of them.

Let's start by building a Torus. It's not strange, but it's useful as an example of interesting gluing. We can make a torus out of simple rectangular manifolds quite easily. We start by building a cylinder by doing the circle construction using rectangles instead of just line segments. So we take four squares; curve each one into a half cylinder, and then overlap them, like in the following picture. The picture shows the
four squares; each of the edges are colored, and each has a dotted line down the middle, which is
where the two matching-color edges of the neighboring rectangles will meet. That gives us a cylinder.

Now, take four cylinders, and using the cylinders instead of line-segments, repeat the same process we used for the circle. So each cylinder is curved into a half-torus, and they're overlapped. And we have a torus, by gluing. (Sorry, but I can't figure out a good way to draw toruses with color bands to show the overlaps using my software!)

Ok.. That was an easy one.

To make some of the harder ones a bit easier, here's a simple trick; if a manifold has edges, those edges can be glued together in the same way as edges of *different* manifolds. This is subject to the exact same kinds of constraints as gluing together multiple manifolds - you need to have reversable
continuous functions for the overlaps.

Using that, we can create a Mobius strip very easily. Take a rectangle, like the following image:

Take the colored edges, and glue them together so that the colors match. To do that, you'll need to put a twist into it. The result is a mobius strip: a manifold with only one edge.

Now, take two mobius strips: one with a right-handed twist, and one with a left-handed twist. Put the two side by side, and glue their edges together. What you'll wind up with is a fascinating shape called *Klein bottle*: a manifold with no boundaries. A true Klein bottle can't really be represented in three dimensions, because any three dimensional embedding will require the manifold to *cross through* itself, which it doesn't do. But an approximate image of a Klein bottle (from Wikipedia) look like:

Another way of making a Klein bottle is to take a square, and glue it so that the edges all match in the following diagram.

What's cool in an extremely geeky way is that there are people who make *almost* Klein bottles out of glass or paper. They're fascinating things - a bottle with *no inside* and *no outside*. There are even [Klein bottle *beer mugs*!](http://www.kleinbottle.com/drinking_mug_klein_bottle.htm). (If any readers ever wanted to make me very happy, just send me one of these. I've never been able to convince myself to spend that much on a mug.)