Is It Possible To Completely Fill a Klein Bottle?

The Klein

bottle

was first imagined by a mathematician named Felix Klein, who wondered what would happen if you took two Mobius strips and sewed them together at the edges. The result is a Klein

bottle

. You can see that if you follow an edge, the outside becomes the inside. According to the topology, the Klein bottle has no volume, but if it has no volume, what happens if we try to

fill

it with water so that when we try to

fill

it we can get up to here and it is already full? Then you have to tilt it inwards. water, it fills it little by little, well, if we just pour a little bit of water in, you'll see that we're not going to be able to get the air out to let the water in, so you can see that you're I always get blocked when trying to pour more in here.

I'm going to try letting the air out and then pouring the water in, but I have a better way to fill a Klein bottle. I'm going to put it in my vacuum chamber and suck out all the air and then when I let the pressure come back in, it will force the water into the vacuum in the bottle. To do this, I will try to stick the Klein bottle down so it doesn't float. I'm going to pour some water around now we're going to try to get that water into the Klein bottle, okay, turn it on three two one so it immediately bubbles because the air is expanding from the Klein bottle and coming out the bottom.

That bubbling comes from the air inside the Klein bottle, so we're trying to make the inside of the Klein bottle as rarefied as

possible

. Okay, let's see what this looks like now when I stop it. Okay, let the air in and it should. suck it here we go, it's already going in, that's amazing, fill it up, I didn't fill it all the way, okay, let's turn the vacuum on again and let this run for a while longer, then okay, here we are. taking this smaller volume of air and rarefying it again, okay, do it again, a Klein bottle almost

completely

full, so I was able to fill a shape with no volume.

Now, at this point, you might be wondering why I would say that the Klein bottle has no volume. It is because to have a volume you have to divide the inner world from the outer world, for example this sphere has a volume, there is a world outside the ball and a world inside the ball and no matter how you stretch or manipulate the shape, there will be There will always be a volume, but let's look at the Klein bottle, where does the outside world end and the inside world begin? There is no fine line or border to divide it into an interior or an exterior.

Now the fact that the Klein bottle has no volume is actually not that special, for example, this glass here has no volume nor is there any division between the inner and outer world. We can artificially define the inside of the cup as anything below this line, but that's just an arbitrary definition of the inside which we are used to living with because we know that when we pour something into it it will stay there due to gravity, but in topology This cup is no different than a flat disc. Now this problem with shapes that have a volume was done for the first time in two dimensions. by mathematician Camille Jordan realized that there is no rigorous mathematical proof to distinguish between a curve that encloses an area like this one or a curve that doesn't like this one.

He then created the first proof of the Jordan Curve Theorem. That is why it is named after it and it is actually one of the most difficult tests that exist, for example, we can look at these two lines and instantly know that they are different because there is one that encloses an area and another that does not, it seems obvious, but To prove it, this is what it takes. Math papers use something called Lemma. A Lemma is an intermediate proof that will be needed later to prove what you are ultimately trying to prove, which is why this article publishes the proof of the Jordan Curve Theorem. shows Lemma 1, Lemma 2, Lemma 3, then Lemma 4 and finally comes to the actual proof.

All of this is to demonstrate what we can see instantly with our eyes, so sometimes there are things obvious to our eyes that are not obvious in mathematics, but some interesting results. This happens once you start to understand these curves, for example a simple closed curve like this means that if you draw a point inside the curve, no matter how you move or stretch the shape, it can't come out. In this example, it's easy to see. that the point is inside the closed curve, but what about this closed curve? The point inside or outside this curve?

Now that we understand these curves, there are some rules we can follow if we draw a line between the point of interest and another point that we know is outside the curve, then you can count how many times it crosses the curved lines and if it crosses it an even number of times it is on the outside, if you cross it an odd number of times it is on the inside, now let's go back to our Klein Bottle from the beginning, notice that I call this a physical Klein bottle because a true Klein bottle doesn't actually exist because it needs four dimensions to be a real Klein bottle, so this is a 3D dive of a 4D Klein bottle, a proper Klein bottle.

The bottle does not intersect itself, but it actually elevates this portion to the fourth dimension, so it does not intersect. You can see in this 3D slice 4D visualizer what it would look like. This is the self-intersecting Klein bottle in 4D vision. 3D slices and this is a non-intersecting Klein bottle, we have an extra dimension so it can move on itself and doesn't have to intersect, so a true Klein bottle can only exist in four dimensions, so in three dimensions. a Klein bottle is just a very fancy cut that needs a vacuum chamber to fill easily and thanks for watching another episode of the action lab.

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