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Trippy shapes

Video transcript

(QuanQuan) If we take a strip of paper like this one, with the blue side and the red side, and we twist it like this, then we make a cylinder. Now, if we take a pencil and we trace it starting from this point, and then go around the cylinder, we see that we only drew on the red side of the cylinder. But let's say we take another strip of paper, again with the red and blue side, and we twist it like this. Now, this shape is called a 'mobius strip'. It kind of looks like a cylinder, except it has a twist in the middle. If we start drawing from the blue side, we take our pencil and go all around the Mobius strip; we see that it crosses into the red side. Now, this is very strange, because for the cylinder, we only traced on the outside, which is the red side of the cylinder, and never touched the inside. But for the Mobius strip, we touched both the red side and the blue side. So what exactly is the outside and the inside of the Mobius strip? Well, the truth is, there is no outside and there is no inside. It's one-sided. The cylinder is an 'orientable surface'. While the Mobius strip is a 'non-orientable surface'. (Jenny) If I wanted to make a Mobius strip scarf, I could just take a rectangular piece of fabric, fold it in on itself while introducing that twist, sort of like what QuanQuan did with that sheet of paper. I would then sew all up the seam, producing a proper Mobius loop. But I could do something even cooler - I could knit a Mobius scarf, starting from its center and introducing the twist when I first cast off. Then, by knitting all along the Mobius strip's one edge, I can gradually widen the scarf to produce a loop without any seam. Behold - a wearable Mobius strip. We can also represent other non-orientable surfaces as models, [struggling] but they're a little bit...harder... to represent. (QuanQuan) As hard as it was for you to take off that scarf? (Jenny) So if we think back to the Mobius strip, we were taking a two-dimensional surface and essentially lifting it into the third dimension and introducing a twist, thus allowing us to create a non-orientable surface. We could theoretically create a three-dimensional non-orientable surface by taking the cylinder - which has length, width, and height - and lifting it into the fourth dimension in order to twist it in on itself. (QuanQuan) I have a hard time picturing it and it looks like you have a hard time making it, too. (Jenny) Well, we do live in a three-dimensional world, so it's kind of hard to go into the fourth dimension to do things like this. We can get pretty good approximations, though. (QuanQuan) Let's take a look at a computer model to see what we're talking about. This bottle is a non-orientable surface and is known as a 'Klein bottle'. The words 'inside' and 'outside' have no meaning for this shape. So we can't make a true Klein bottle, but we can get pretty close. So Chris over here is an MIT senior and one of the founders of NVBots, and he's going to help me 3D print one. This printer takes this spool of plastic, runs it through this device called an 'extruder', which operates like a hot glue gun that melts the plastic so it can print the bottle layer by layer. So this is actually a pretty good representation of the Klein bottle. If you look at the bottom half of this bottle, you can see that the outer surface loops back into the inner surface so that there really is no outer or inner surface. There's only one side to this bottle, just like the one side that's in the Mobius strip. But if you look at the top half of this bottle, the neck comes out and goes back into the bottle through a physical hole. In a true Klein bottle, this hole doesn't exist because the neck comes out into the fourth dimension and then loops back in and the bottle is connected the whole way through. (Jenny) Thank you.