In this series of articles, slowly but surely, we uncover the secrets of the fourth dimension. We already got familiar with the hyper-cube, particularly the tesseract – a fourth-dimensional variation of the cube. Although cubes are great, and fourth-dimensional ones are even more than others, you have to agree with me on this one: as fascinating as they are, they are still quite simplistic. Now it’s time to deal with a more sophisticated fourth-dimensional shape! A shape with some untrivial, and even unintuitive properties: The Klein bottle.
To do so, we have to start slowly, gradually, with something basic, like a square:
Now we’ll take this square and start gluing it up: take one pair of parallel sides, and glue them together, so in the resulting shape, if you travel through one of them, you’ll appear on the opposing side. Can you imagine what shape we’ve constructed? A cylinder!
But what if one of the arrows was directed in the opposite direction, i.e., up instead of down? If you had to glue the arrows in a way that once they are glued together, they turn to the same side, what shape you’ll get?
The resulting shape inspired people over many centuries, and it’s called a Möbius strip:
You can easily create one of your own: take a long piece of paper, or a ribbon, give one end a half-twist (180 degrees) and connect the ends together. It’s as simple as that. To be fair (or precise), it’s not really a Möbius strip but a representation of one, because the real Möbius strip is two-dimensional and has no width, while a piece of paper is three-dimensional with width.
In any case, you may wonder, what’s so inspiring in it? After all, it’s just a ribbon with a (half-) twist. The thing is that it has only one side. How come? If you take a cylinder, you clearly can mark its two sides: paint one of them in one color and the other in another. But you can’t do that with the Möbius strip. If you try coloring one of its sides, you’ll see that you have covered the whole shape with the same color. In the same way, the ball in the image above rolls on what at first glance seems to be the “internal” and “external” parts of the shape, without switching sides.
But we are here for more complicated shapes, and we want them to be in 4D! Now, here’s a challenge: can you imagine what you’ll get if you glue together two Möbius strips edge-to-edge? We’ll leave this question open for a while. Here’s another one: what shape you’ll get if you glue the arrows of this square (connect the arrows with the same color and direction):
That’s relatively easy: first, you have to choose one color and connect it together, and you’ll get a cylinder, as we saw before. Now, connect the two remaining arrows, and get a torus (a donut-like shape):
Now, what if one set of arrows was pointing in the opposite direction?
We already know that this makes things more complicated, twisting the resulting shape, and that’s exactly what happens here: similar to the way that the Möbius strip is like a cylinder but with a half-twist, the resulting shape is like a torus, but twisted:
It’s called a Klein bottle, and in the same way as the Möbius strip is a two-dimensional shape embedded in a three-dimensional world, the Klein bottle is a two-dimensional shape embedded in a fourth-dimensional world.
By Tttrung – Own work, CC BY-SA 3.0
When we look at the Klein bottle in our three-dimensional world, we see it as self-intersecting – that doesn’t happen in the fourth dimension. In it, it doesn’t self-intersect. Also, like the Möbius strip, the Klein bottle has only one side. I.e., it doesn’t have an inside or an outside. Hence, no volume. Take your time to validate it in the image above.
Now back to the question about gluing together two Möbius strips: maybe you’ve already figured out that the shape you’ll get is, yes indeed, a Klein bottle:
Things just got really weird. Leo Moser, who was an Austrian-Canadian mathematician (1921-1970), described it in the following limerick:
“A mathematician named Klein
Thought the Möbius band was divine.
Said he: “If you glue
The edges of two,
You’ll get a weird bottle like mine.”
Featured image: By Alex Healing – Spiral Klein Bottle, CC BY 2.0
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