I’ve heard about this idea a lot, but on the flip side it doesn’t seem to make sense. Us as 3D beings can’t naturally encounter 2D things, just representations of them (drawings etc), but nothing in the real world that has just 2 dimensions. So how could it be possible for there to be 4 (or more) dimensions?
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You are discounting pretty quickly our ability to encounter 2D things. A drawing of a ball is a representation of a 3D object. That representation is not however 3D. We can create an illusion where it appears to be 3D, but it isn’t. Its a drawing on paper, and that drawing has no actual depth. It being a drawing doesn’t make it any less 2D. However our ability to create an illusion of being 3D is only possible because we understand that there is a 3rd dimension, and can observe it. You can look up “Lego Army” by Leon Keer for an example of this. When we move from the forced perspective, the illusion of a 3D army marching down the street is lost. A 2D person in a 2D universe could never see both perspectives.
Now as for higher spatial dimensions. We aren’t 4th dimensional beings and so while we can certainly describe the idea of a 4th dimensional object, they would have properties that are impossible. The tesseract for example is a 4th dimensional cube. If you imagine a regular cube, you can unfold it into 6 squares, a net, that can then be folded back up to make a cube. If you replaced each of those squares with a cube, and then folded them together, you’d have a tesseract. The problem, of course, is this is impossible. Each cube already inhabits three spatial dimensions, unlike the square net’s two. With the square net we can force it to inhabit a third dimension to form a cube. We can’t force the cube net to form a tesseract. If a tesseract actually existed, from one angle it would appear to be a perfectly normal cube. But the second we moved, its shape would change in the same way that walking around “Lego Army” changes how we perceive it.
So if we don’t have the capacity to perceive 4 dimensions, and we don’t live in a universe where its possible, as far as we know, for there to be 4 dimensions, why bother? Well, the answer as always is Math. Imagine we are a store owner who wants to maximize profit of a particular product. He discovers that there are 4 factors that affect the profit of the product. The price, the time of year, the location in the store, and the time of day. He also figures out how much each variable affects the profit. If he then wants to model this, we quickly run into a problem. This math problem is 4th dimensional. It has 4 axis that any point could be, and so can’t be traditionally graphed. But the math to maximize this problem is very possible without graphing.
This idea of a simple math problem that can’t be physically represented in its purest form led people to want to try to do so and 4th dimensional geometry was born. Just like a 2D person could never see all of the drawing there in the same way we can, we can’t observe a 4D object. But we can imagine and create some very cool ideas.
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