# 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.)

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I feel compelled that's not just "people" you linked to there.

I quote from the FAQ:

Who's behind Acme Klein Bottle?

Just me, Cliff Stoll. Nobody else. No management. No employees. No investors. No profits, either.

Are you the same guy that ...

Yep, same guy.

By Pseudonym (not verified) on 08 Nov 2006 #permalink

One Christmas I decided to get my wife a Klein bottle, so I called all the local glassblowers I could find. I fully expected to have a terrible time explaining what the hell I meant, but it seems the (almost) Klein bottle is a standard teaching tool in glassblowing school. Every one of the artists knew what I meant. Not all of them worked in borosilicate, but those who did were all capable of making a Klein bottle. I paid about \$120 for a 10" tall one.

By Johnny Vector (not verified) on 08 Nov 2006 #permalink

Can you obtain a Klein bottle by gluing together the two edges of a Mobius string, too?

I eagerly await word from Karl Rove on what happened to "The Math."

By Mustafa Mond, FCD (not verified) on 08 Nov 2006 #permalink

If you're ever in London, the Science Museum has a quite nice little collection of blown glass klein bottles and variants, in a display case - in the mathematics section, of course.