I can’t comprehend it
In: Mathematics
It’s kind of the 3D version of a Mobius strip.
A Mobius strip is 2D in the sense that it’s a flat surface and if you’re *on* the strip you can go as far forward/backward as you like (along the strip) and cover the entire surface (both sides) but there’s an edge to the left or right you can’t cross. You can keep going forward forever because the strip curves around in 3D to rejoin itself.
The Klein bottle is a 3D shape that curves around in 4D to rejoin itself. There’s a 3D version (you can buy them, they’re fun, I personally love Acme Klein Bottle [https://www.kleinbottle.com/](https://www.kleinbottle.com/)). As you can see, in 3D the bottle passes through its own surface, which kind of feels like cheating…because we can’t do 4D in our 3D universe. If you could do 4D, which mathematicians can, the bottle would have all the same properties but wouldn’t intersect itself.
NOTE: this explanation assumes the reader knows what a Klein bottle is.
It has to exist in 4 spatial dimensions. If it is just 3 dimensional, then the neck has to intersect the body of the bottle. By definition, the bottle can’t intersect itself, so in order for the surface to be completely continuous, it has to ‘jump’ over the 3D surface in the direction of a 4th dimension.
I don’t know if that made any sense — it’s a difficult thing to explain with just text. Think about a 2D figure ‘8’. In 2D, the middle of the 8 intersects itself. But you could imagine it in 3D as a string, where the string jumps over itself in the 3rd dimension. The same thing is happening with the ‘intersection’ in the Klein bottle, only it jumps over itself in a 4th dimension.
Let’s think with portals!
[Imagine a Klein bottle like this instead](https://i.imgur.com/V3WnGpX.png). The portals mean that the outer surface is completely smooth even though the tube connects through it.
Now imagine instead of portals, the tube and outer surface cross in the 4th dimension. So that, like with portals, the outer surface is completely smooth and unbroken, and the tube is complete but also goes through it.
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