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In zero dimensional space, differentiation is impossible. It's necessary (as far as we know) for things to be in two different places to distinguish between them. So, as Euclid says, a point is that which has no part. There are no parts to be had and no difference between said non-existencent parts.

In one dimensional space, is it possible for an observer to compare any objects? Are two dimensions necessary for measurement of distance and extension? In 1D translation is possible but rotation requires 2D to be meaningful.

What could one do in three dimensional space that one couldn't do in two dimensions? In 4D, we could see inside of things, but so what? Wouldn't it be more of the same? So, I can see why one and two dimensions may be essential to any kind of emperical existence we might be able to relate to but I fail to see how three or more dimensions would be any "better" than two.

Hawking once said that in Flatland the inhabitants wouldn't be able to digest their food and there's a preface in the book talking about them having a height of which they aren't aware of but for this question I'd like to assume that point particles can interact in a classical kind of way (with time) because I'm more interested in a functional perspective: what do we get as observers from being in 2D, rather than 1D, for example?

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    A classical paper from 100 years ago: P. Ehrenfest, In that way does it become manifest in the fundamental laws of physics that space has three dimensions? in: KNAW, Proceedings, 20 I, 1918, Amsterdam, 1918, pp. 200-209 dwc.knaw.nl/DL/publications/PU00012213.pdf
    – sand1
    Jul 30, 2019 at 20:24
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    No object could exist in Flatland. It's a fun idea but to suppose an object could exist in two dimensions is to suppose they have a height. Hence the comment about them having a height of which they aren't aware.
    – user20253
    Jul 31, 2019 at 11:24
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    @PeterJ I think that to suppose an object could exist in two dimensions is rather to suppose that they don't have a height. I'm also quite convinced that most objects have no awareness to speak of.
    – jhch
    Jul 31, 2019 at 18:21
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    Our definitions of object differ.
    – user40425
    Aug 1, 2019 at 12:49
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    There are objects with asymmetries that can be inverted in 3-space but not in 2-space. For example, I can "flip" an "R" in 3-space to get its mirror image, but I can only "rotate" an "R" in 2-space, and so I cannot get its mirror image.
    – nwr
    Aug 2, 2019 at 2:36

3 Answers 3

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It's a bit unclear what you're asking, but I'll run with the part: "What do we get as observers from being in 2D rather than 1D?" I've also italicized a few things which answer the sub-questions in your question.

TL;DR You get a lot of complexity with each added dimension, and that might seem nice for humans. But lots of cool things happen in just a few dimensions. For example, sound is a two dimensional phenomenon. Maybe we wouldn't be able to process it (brains are at least three dimensional), but music could exist nonetheless and I think there's something to be said for that.

Let's start with Dimension 1:

It's easiest to think of this as a line. We don't get much with just one dimension, and the world is pretty boring. We can think of distance between points on that line, and we can think of whether or not points are on top of each other (distance of 0). With that, we can say "I'm further away from point A than point B, and if I was closer to point A I would be further from point B."

There's another way of thinking of one spatial dimension. Imagine a single point, which we assign a "direction." What does that mean? Well... It can point any "direction" we want, from 0 to 360 degrees. That's no different than assigning points on a number line. The "angle", as we call it, is the dimension. And we can talk about how two angles are different, or are the same. In fact, we usually say that angle 0 and angle 360 are identical, which gives some weird properties. "I'm further away from angle A than angle B, but if I rotate closer to angle A I also get closer to angle B in some way." Funky! But this lets us talk about rotation in as few as one dimension.

But obviously, there still isn't that much going on in one spatial dimension.

Dimension 2:

It's easiest to think of this as a plane, like a map. There is an up/down and a left/right. Or you can think of it as latitude and longitude, which is sorta like a map except all the edges are connected. Or you can think of it as polar coordinates, like a radar readout.

What distinguishes two dimensions is that two points can be in the same place in one dimension but different places in the other. We can do a lot more with two dimensions. Distance becomes much harder to define. Interesting shapes emerge. The world as you would see it through one eye, and hear it through one ear, begins to exist. Flat drawings can exist, although I'm not exactly sure how one could perceive them (if I recall correctly, Flatland really gets at this).

Sidebar on sound, because I think it's fascinating: Against your eardrum is a wall of air (a 2D plane), which moves in a direction perpendicular to the plane (a 3rd dimension), over a period of time (a 4th dimension) to make you hear noise. However, sound is really a two-dimensional phenomenon! A microphone converts our many-dimensional sound into a two-dimensional signal, and a speaker converts the signal back into four-dimensional sound. The signal in the middle is still "sound" in a pure sense--it's just that humans aren't designed to hear in two dimensions so we need to it be converted up. So when it comes to noise, there is no benefit to 3 or 4 dimensions except that humans are designed to work in that number.

Lets get back to...

Dimension 3:

This is where things start to get really real. Each dimension, left/right, up/down, in/out, is independent of the others. The world, in a snapshot, exists. Sculptures can exist. You, geometrically speaking, exist. Textures exist. There are a whole variety of three-dimensional spaces, which can take on strange shapes and properties. But even in our own pretty much Euclidean space, three dimensions lets us easily perceive the vast breadth of two-dimensional objects with our eyes and our ears. A two dimensional drawing could exist in two dimensions, but to actually see it, you need three.

But we're still kinda missing something...

Dimension 4?

Time (usually). This not a spatial dimension, but it's important. It gives us before and after, movement, etc. It gets pretty funky once you look at special or general relativity, but that seems out of the scope of this question. What matters most is that it animates the other spatial dimensions. But it doesn't have to be the fourth dimension. You could just as well have a world with only two dimensions: a line, and time. Or three dimensions: Cartesian space, and time, kind of like Flatland.

What I hope to show by this is that with each added dimension, we get a slightly more complex world which we perceive to be beneficial to humans purely because of the way humans perceive the world.

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to know the benefits of three-dimensional space we have to first understand the problems which we would have faced by not knowing about three-dimensional space is. Some of them are:-

  1. We would have never known what is the difference in reality because we would have never had the opportunity to look at things as they were. Let's take an example of the view that we get in a 2d film at the theatre and the view that we get for a 3d film inside a theatre. So 3d picture seems as to us that the object or the event is happening in front of our eyes as if it was real.

  2. We would have never known the true meaning of perspective because we would have always had the same picture irrespective of how many times we would have looked at it. So this would have led us to a partial reality where we would have never seen things in front of our eyes but as if we were looking at a monitor.

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Atoms have finite size in three dimensions. Therefore, for atoms to exist, you need three dimensions of space, which means that for molecules to exist, you need three dimensions. Since life as we know it is based on atoms and molecules, you need three dimensions for life as we know it to exist.

Why not four dimensions for space instead? To begin with, gravity in four spatial dimensions mathematically yields no stable orbits- meaning no orbiting moons, no solar systems, no galaxies, no galaxy clusters, no superclusters. Similarly, electrostatics in 4D yields no stable orbitals for electrons near nuclei, meaning no atoms, no molecules, and hence no life as we know it. In fact, this fundamental instability is present in any universe with an even number of extended spatial dimensions bigger than two.