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Mindbending Things about the 4th Dimension

We live in the 3rd dimension and are trying to imagine a 4th dimension. You might already know that time is a 4th dimension in our universe, but what I’m talking about is a fourth dimension of space, which, including time, would technically be five dimensions.


It’s pretty difficult to wrap your head around four spatial dimensions, but there’s an analogy you can use to help understand. Imagine a two dimensional person, only able to move up, down, left, right, but not forward or backward. Imagine this person is trapped in a box with four walls. From our perspective we can see the walls, the person inside the box, and the stuff outside of the box at the same time. The 2d person can only see the walls inside the box, and to them the box is just entirely surrounding them. 


From our perspective the walls have no width, you could cut your finger on them. If you were the one trapped in this 2d box, you could just step out to escape. In a four dimensional world, a 3d box wouldn’t have any 4th dimensional width, the same way a 2d box doesn’t have any three dimensional width. If a 4d person was trapped in a 3d room with no exit, they could just step out. Also, a 4d person could get a papercut from a 3d cube the same way we get papercuts from paper.


If you didn’t fully get it, that’s fine, the 4th dimension is pretty hard to understand. I will now talk about an object that you can make in real life. You can follow along if you have a long strip of paper. Take the strip of paper, and connect one end with the other end. This should just give you a loop, not very interesting. If you do the exact same thing, but twist it before connecting it with itself, you get something much more interesting, called a Mobius strip, like the image above. The interesting thing about this object is that if you take a pencil, and start drawing a line around it, you can draw on every side of it without lifting your pencil or going around an edge. It’s an object with one side. 


Now, take a second Mobius strip, and connect both of them together along their edges. You’ll end up with something that still only has one side, but this time it doesn’t have any edges. If you try to do this though, no matter how hard you try you won’t be able to connect them together. It’ll just bump into itself when you try to connect the edges. This is because our three dimensions aren’t enough to let it move past itself. This object is actually possible to make in four dimensions, and mathematicians call it a Klein bottle. The image to the left is almost a Klein bottle, but since we live in three dimensions, there had to be an intersection. Just like the Mobius strip, if you take a pencil and draw along the Klein bottle, you can get to any side of it. You could even color in every side of the Klein bottle without lifting your pencil.


There’s a lot of other interesting things about the 4th dimension. This next thing might be hard to understand if you aren’t geometrically minded. Imagine a 1x1 square sitting diagonally inside a 2x2 square so its points are pointing towards the edges. It doesn’t touch any of the edges since it’s too small. Imagine the same thing, but with a 1x1 cube inside a 2x2 cube. The tips of the inside cube still don’t touch the outside cube, but they’re closer. If you did the same thing with a 1x1 four dimensional cube, inside a 2x2 one, it would actually touch the edges of the 2x2 one. If you really want to make your brain hurt, a 5d cube actually wouldn’t even fit if you put it diagonally.


You might think this doesn’t have any application in the real world, and you’re mostly right, but it’s still just interesting to think about. It does actually have some applications, too. Some theories about how our universe works involve there being as many as 11 dimensions, and if our universe actually works that way, thinking about this stuff could lead to a deeper understanding of our universe.

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