When the GIF starts, the dots listed off clockwise are black, blue and red. This transformation is impossible on an orientable surface like the two-sided loop. The concept of orientability has important implications. Take enantiomers. These chemical compounds have the same chemical structures except for one key difference: They are mirror images of one another. For example, the chemical L-methamphetamine is an ingredient in Vicks Vapor Inhalers.
Its mirror image, D-methamphetamine, is a Class A illegal drug. This is easy to do, and easy to visualise. But this article is going to be an introduction to topology. In topology we discard the concept of distance and angle. We are allowed to smoosh things about, stretching, squashing, and generally deforming things. No tearing, cutting, and glueing allowed, unless we put things back afterwards.
Topology does talk about shapes, but it turns out that that description is attractive, catchy, and wrong. We will see why later. So we take a fresh look at the cylinder, and ask a rather odd question: How many sides does a cylinder have?
In geometry, a square has four sides, but we might equally well say it has two sides — front and back. Indeed, if you want to paint a square you can use two different colours without the colours meeting each other. Of course, this article is about topology and not geometry , and in topology a square is the same as a triangle, which is is the same as a disc. And that is the concept we use. But examples are the best way to make this clear. In topology, a cylinder has two sides inside and outside and two edges.
A disc, by comparison, has two sides, but only one edge. We can run our finger all the way around the edge of a disc without ever lifting it off, and we cover all parts of the edge. So a cylinder has two sides, and two edges, while a disc has two sides, and only one edge. These facts remain true, even if we then distort the cylinder or disc — bending, stretching, twisting — we still have the same number of sides, and the same number of edges.
Now take another strip, bend it around, align the short edges, and before joining them, insert a half twist. When first presented to people it can cause quite a stir, especially when you ask them to cut it in half. Instead we get one long strip with some number of twists in it.
Cutting it in thirds is even more entertaining, because then we do get two pieces, but they are different sizes, and they are linked! In fact, with a small amount of analysis some of these facts can be deduced in advance. We call this surgery — we cut things into pieces, but remember which bits need to be sewn back together.
So here we cut the strip, then untwist, and open it out. In the resulting diagram the two ends should be regarded as, in principle, still joined, with the arrows showing that we need to have the half twist. The locations marked and are still quite close to each other. Cutting it in half again makes two interlocking loops.
Cutting each of these loops again will make four interlocking loops. Try it with three or more twists. The loop of paper with a twist was first described by a German mathematician called Johann Benedict Listing. Next we bend the cylinder and then glue together the ends so that the arrows agree.
We cannot realize this figure in three dimensions. We can pretend that we pass the end of the cylinder through itself before gluing the ends together, but mathematicians often prefer to think of this object as existing in 4-dimensional space. Just like the Mobius stip, the Klein Bottle is a one-sided figure. Unlike the Mobius Strip, the Klein bottle does not have any boundary though.
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