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File: MathsInATwist [[[> A MathematicsTalk given _ by ColinWright _ _ TaggedAsTalkDescription ]]] Many students are introduced to the idea of the MoebiusStrip, that wonderfully perplexing strip with a half twist that has only one side and one edge, and which when cut in half doesn't do what you might expect. In this workshop we don't just stop there, but explore what happens with other possible twists and turns, and try to find some way of understanding how this works, what else is possible, and whether we can make sense of it all. [[[> | Sides -> _ Edges | |>> 1 <<| | |>> 2 <<| | | |>> 0 <<| | |>> ? <<| | |>> ? <<| | | |>> 1 <<| | MoebiusStrip | |>> Disk <<| | | |>> 2 <<| | |>> ? <<| | |>> Cylinder <<| | | |>> 3 <<| | |>> ? <<| | |>> ? <<| | | |>> 4 <<| | |>> ? <<| | |>> ? <<| | | |>> ... <<| | |>> ... <<| | |>> ... <<| | ]]] We start with the MoebiusStrip and the Cylinder. Looking at these carefully we can see that one has two edges and two sides (being careful to define what we mean by "edge" and "side") while the other has only one edge and one side. When we cut the cylinder in half around its waist we end up with two things, both basically cylinders, but half the height. That lets us talk about whether the actual size matters when we're talking about things. This is topology - so no, the exact size doesn't matter. It's the /form/ that matters, whatever that means. Because of that we can take a cylinder and stretch the bottom edge out until we get a lamp-shade shape. Then we can squash it flat until we get an annulus. In other words, a cylinder is the same - topologically - as a disk with a hole. So a disk has one edge and two sides, a cylinder is a disk with a hole, so that's two edges and two sides. Maybe going down the chart is simply adding holes to things. Is it? And if going down the table is /adding/ holes, then going up the table should be /subtracting/ holes, but what is a disk minus a hole? Does that make sense? [[[>50 For this workshop everyone needs paper, pencil or pen. In addition, everyone needs access to scissors, and sellotape - one between two can be made to work, but one each is better. ]]] When we cut the MoebiusStrip in half around its waist we /don't/ get two pieces, we get just one. The MoebiusStrip and the cylinder differ in some fundamental way. That's what Topology explores. Then we go back to the number of sides and the number of edges, and see if we can start to find some sort of pattern. Can we find a way to fill the blanks? Do we always get what we expect? ---- This is one of the MathematicsTalks offered by ColinWright.