I just watched [this video](https://youtu.be/9Bqg-6nzkzw) by the action lab, and he explained, vaguely, how a klein bottle would work in the 4th dimension, but my little rat brain doesn’t understand one bit of it.
How could the neck part of the bottle not intersect the wall of the bulb?
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We, sadly, do not live in four dimensions. We live in three. The object in that video is not, in fact, a Klein bottle. A Klein bottle is a three-dimensional object, but it **cannot exist** in three dimensions; this is the same as how a Mobius strip is a two-dimensional object, but it cannot exist in two dimensions, but only in three. So what is the Action Lab guy actually holding? He’s holding the shadow (the math term would be “projection”) that a Klein bottle makes in three dimensions.
Think about you and your shadow. You’re a 3D object, the ground is a 2D object. The shadow you cast on the ground is 2D. Your shadow closely resembles you, but it is different. Simpler. Imagine facing with your back to the sun, and putting your hand behind your head, but not touching. From your shadow, it looks like your hand *is* touching your head, but in the 3D space you actually inhabit, it isn’t. This is an example of the sort of detail that’s lost when you project down to a lower dimension.
In the 4-dimensional space that a real Klein bottle exists in, it does not self-intersect. The geometry is fundamentally different, and there’s an extra dimension to work with so that you can bring the mouth of the tube to the bottom of the bottle from the inside, without having to go through the wall.
Consider a Mobius strip. It must exist in 3D. From 3D, we see that the two edges of the Mobius strip never touch each other; they are always separated by the same distance, the width of the strip. But consider the shadow: there is no way to make a 2D shadow of a Mobius strip so that the edges don’t touch. I recommend you try it! Take a piece of letter/A4 paper; cut a strip 1″ (2-3cm) wide, put a half-twist in it, and tape the edges together. Now find a bright light in your house that casts a good shadow, and play around, seeing if you can find a shadow that doesn’t have the edges end up crossing each other.
The Mobius strip is a 2D object in a 3D world, and taking the 2D shadow causes it to lose the ability to not self-intersect. A Klein bottle is a 3D object in a 4D world, and taking the 3D shadow causes it to lose that same ability to not self-intersect. And remember, what we see isn’t the Klein bottle itself, but rather its shadow.
Does that help at all?
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