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Trippy Shapes

5 minutes

(Describer) Science Out Loud.

If we take a strip of paper like this one and we twist it like this, then we make a cylinder.

(Describer) Red end meeting red end.

If we take a pencil and we trace it starting from this point, and go around the cylinder, we see that we only drew on the cylinder's red side. Let's take another strip of paper with red and blue sides and twist it like this.

(Describer) Red end meeting blue end.

This shape is called a Möbius strip. It looks like a cylinder, except it has a twist in the middle. If we start drawing from the blue side and take our pencil and go all around the Möbius strip, we see that it crosses into the red side. This is strange, because for the cylinder, we only traced on the outside, the red side, and never touched the inside. For the Möbius strip, we touched both sides. What is the outside and the inside of the Möbius strip?

(Describer) Question marks appear.

The truth is, there is no outside and there is no inside. It's one-sided. The cylinder is an orientable surface, while the Möbius strip is a non-orientable surface.

(Describer) Another woman holds a scarf.

If I wanted to make a Möbius scarf, I could take a rectangular piece of fabric, and fold it in on itself while introducing that twist, like QuanQuan did with that paper. I'd sew all up the seam, producing a proper Möbius loop. But I could do something even cooler. I could knit a Möbius scarf, starting from its center and introducing the twist when I first cast off. By knitting all along the Möbius strip's one edge, I can gradually widen the scarf to produce a loop without any seam.

(Describer) She knits and wears it.

Behold, a wearable Möbius strip. We can also represent other non-orientable surfaces as models, but they're a little bit

(Describer) She pulls the scarf over her head.

harder to represent. As hard as it was for you to take off that scarf?

(Describer) The second woman frowns at Kwan Kwan, then holds some paper.

If we think back to the Möbius 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 this cylinder-- which has length, width, and height-- and lifting it into the fourth dimension in order to twist it in on itself. I have a hard time picturing it. It looks hard to make too. We live in a three-dimensional world. It's hard to do things in the fourth dimension. We can get good approximations, though. Let's view a computer model to see what we're talking about.

(Describer) The neck of a bottle opens at the bottom of it.

(QuanQuan) This bottle is a non-orientable surface known as a Klein bottle. The words "inside" and "outside" have no meaning for this shape.

(Describer) A 3D printer creates the shape with holes in it.

We can't make a true Klein bottle, but we can get close. Chris here is a M.I.T. senior, and he'll help me 3-D print one. This printer takes this spool of plastic, runs it through this device called the extruder, which operates like a hot glue gun that melts the plastic so it can print the bottle layer by layer.

(Describer) When it finishes the shape, Kwan Kwan picks it up.

So, this is a good representation of the Klein bottle. If you look at its bottom half, 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 Möbius strip. If you look at its top half, 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.

(Describer) Sitting at a table with the other woman, Kwan-Kwan sets the Klein Bottle on the table. Then she moves it across the table to her.

(Describer) The other woman picks up the small plastic bottle, and sets it on her head.

Thank you.

(Describer) Titles: Science Out Loud. Made with love at MIT. Accessibility provided by the U.S. Department of Education.

Accessibility provided by the U.S. Department of Education.

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What's a trippy shape? It defies the normal principles of geometry and goes beyond three dimensions. MIT scientists explain, knit, and 3D-print their way through trippy shapes. Part of the "Science Out Loud" series.

Media Details

Runtime: 5 minutes

Science Out Loud
Episode 1
4 minutes
Grade Level: 9 - 12
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Episode 2
6 minutes
Grade Level: 10 - 12
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Episode 3
5 minutes
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Episode 4
4 minutes
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Episode 5
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Episode 6
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Episode 7
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Episode 8
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Episode 8
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Episode 10
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