Higher spacial dimensions

[Just thinking]

I've read and heard that in order for certain theories to be true there must be more than 3 spacial dimensions (11 for String Theory, if I recall) -- but that most collapse well below our ability to measure them, and all we're left with are three. But is it possible that higher order spacial dimensions may be among us in other ways?

An analogy: The surface of a sphere is clearly 3-dimensional. But if the surface is large enough one can travel on it and not realize more than the 2 dimensions like that of a flat surface. Unless the methods of measurement are capable of doing it and one proposes the theory, realizing that one is on a curved 3D surface instead of a flat 2D surface may be daunting.

My question: Is it possible that 4D spacial measurement/detection may be hidden from us in a similar way as the curvature of a sphere's surface was to those of the past? Is it possible that we indeed travel in 4 (or more) spacial dimensions all the time without realizing it due to its relative subtleness?

 

[Jarom]

You state that the surface of a sphere is "clearly" 3-dimensional, but I submit that it is "clearly" 2-dimensional. There's obviously a definitional problem here. The dimension of a manifold (which is, to be a bit imprecise, a space on which it's possible to talk meaningfully about dimension) is (more imprecision ahead) the number of independent directions you can travel from any point. On the surface of a sphere, that's two. In our universe, it's three (ignoring time, which isn't really a good idea, but we'll do it anyway)

Like the surface of a sphere, though, our 3-dimensional universe isn't flat. It's distorted by the influence of mass called gravity. If we wanted to picture our universe as being a 3-dimensional "sheet" inside a flat space, this flat space would need to be have more dimensions. I think four would suffice, but I'm unable to come up with a good reason why. However, it's not clear if this is a useful picture for any purpose.

I hope I helped you more than I confused.

 

[Just thinking]

I hope I helped you more than I confused.

Yes, you have helped.

When I said that the surface of a sphere is clearly 3-dimensional, I meant that it exists in 3 dimensions, or curves into 3 dimensions. One cannot have a sphere (as we define it) in a 2D universe.

Also, when walking on a sphere's surface you do in fact travel in 3 dimensions -- even by just remaining on the surface. You are limited to left and right -- but travel far enough and you do travel up and down. You just can't detect it if the curvature is large enough and you haven't the concept or equipment to help you.

 

[epepke]

Yes, you have helped.

When I said that the surface of a sphere is clearly 3-dimensional, I meant that it exists in 3 dimensions, or curves into 3 dimensions. One cannot have a sphere (as we define it) in a 2D universe.

Well, there are certain topological conventions to this.

We say that the surface of a sphere is 2-D, but that the spheres we know about are embedded in 3-D space.

The essence of topology is to concentrate on only the intrinsic qualities of the space, that is, the qualities that don't require embedding.

Ultimately, no matter how many dimensions you think there are, there has to be a point where you have to ignore embedding, because otherwise you have to have an infinite number of dimensions.

 

[Just thinking]

Ultimately, no matter how many dimensions you think there are, there has to be a point where you have to ignore embedding, because otherwise you have to have an infinite number of dimensions.

Maybe there are ... but that's another topic.

I'm just wondering if like the 3D person walking along the surface of a sphere (who believes he's only traveling in 2 dimensions but is actually moving in 3) we constantly move in more spacial dimensions (4?) than we actually think we do (3).

 

[epepke]

Maybe there are ... but that's another topic.

I'm just wondering if like the 3D person walking along the surface of a sphere (who believes he's only traveling in 2 dimensions but is actually moving in 3) we constantly move in more spacial dimensions (4?) than we actually think we do (3).

I can't tell. This would require some 4-D observer. I can actually grok four spatial dimensions, but this is only because I got to see a 3-D projection of some 4-D objects illuminated with 4-D light in the Cave.

However, there are good reasons (at least, they seem good to me) that realities can only have certain numbers of spatial dimensions, namely 0, 1, 3, 7, 15, 31, and there ain't no more.

 

[Schneibster]

You state that the surface of a sphere is "clearly" 3-dimensional, but I submit that it is "clearly" 2-dimensional. There's obviously a definitional problem here. The dimension of a manifold (which is, to be a bit imprecise, a space on which it's possible to talk meaningfully about dimension) is (more imprecision ahead) the number of independent directions you can travel from any point. On the surface of a sphere, that's two. In our universe, it's three (ignoring time, which isn't really a good idea, but we'll do it anyway)

You guys got this one right, it's a 2D surface bent in the third dimension.

Like the surface of a sphere, though, our 3-dimensional universe isn't flat. It's distorted by the influence of mass called gravity. If we wanted to picture our universe as being a 3-dimensional "sheet" inside a flat space, this flat space would need to be have more dimensions. I think four would suffice, but I'm unable to come up with a good reason why.

I can help. Your problem is that you're envisioning adding a third dimension to two that already exist (or a fourth to three) rather than combining it with them. Also, you're stuck on a confusing point about the difference between an axis of rotation and a dimension, and this will cause you a great deal of trouble in envisioning it. But with the correct way to think about it, you'll find it's much easier. I'll see if I can tell you that way.

Let's begin with one dimension. Movement is possible in only one direction. Rotation is impossible; there is no direction in which to rotate.

Now, let's combine a second dimension with this first one we started with. Now, we can move not merely in one direction, and not merely in two, but in an INFINITE number of directions. By merely adding one dimension to the one we already had, we have not DOUBLED our possibilities, we have INCREASED THEM INFINITELY. This will not be the last time that happens. But we've done something else. In two dimensions, we can do something new: objects can rotate. So we've found something out: the minimum number of dimensions required to support rotation is two.

There's something else I want you to notice. What is the direction of the AXIS OF ROTATION? Uh-oh; there ISN'T ONE. It sticks out in a direction that doesn't exist; remember, we only have two dimensions. And we've discovered something else again: "axis of rotation" is a meaningless term in the simplest system in which rotation can occur. So how can we define the direction in which rotation can occur? Simple: PLANE of rotation. And we will never refer to an axis of rotation again, because we now know that it's meaningless.

Now, how many planes of rotation are there in two dimensions? Simple. One. That's it. There's only one direction you can rotate in, and it's defined by those two dimensions. Anything in our two dimensions that's rotating, is rotating in that plane, period.

OK, now let's combine a third dimension with those two we already had. What do we gain? Well, first, we now have added two more infinities of directions in which we can move objects. I told you this would not be the last time that happened. We've done it again. What's more, we've MORE than done it; we've added an INFINITY OF INFINITIES of directions we can move objects in. OK, now what about rotations? We combined in one more dimension; how many planes of rotation do we have now?

Interestingly, adding ONE dimension to ONE dimension produced ONE plane of rotation; but adding ONE MORE dimension to that two didn't just add ONE more plane of rotation; it added TWO more. Now we have three planes of rotation.

Now, let's take just one more look at our "axis of rotation" concept before we discard it forever. We'll do this so that we can understand precisely why it confused our understanding. Note that the axis of rotation of an object rotating in one of our three available planes of rotation ALWAYS POINTS IN THE DIRECTION OF THE DIMENSION THAT IS NOT PART OF THE PLANE OF ROTATION. So whenever we try to envision extra dimensions, the FIRST THING WE DO is try to imagine the extra AXES of rotation. We now see that that's WRONG. We should be thinking about the extra PLANES of rotation, not the extra AXES. Those axes might point in directions that DON'T EXIST in our dimensional framework; so they'll do nothing but confuse us. But the PLANES will ALWAYS exist as combinations of two of our dimensions, and in fact we can always count how many planes of rotations there are in a space by counting the combinations of its dimensions in pairs; however many possible combinations there are, that's how many ways things can rotate.

So let's name our dimensions. We'll call the one we started with "x." The second one we added, we'll call "y." So the single plane of rotation we had in our two-dimensional framework, what should we call that? How does "x-y" grab you? Looks pretty good to me. OK, what shall we call the third dimension? "Z," of course. And the two new planes? "X-z" and "y-z" sound pretty good to me.

OK, now we take the step that takes us beyond our normal abilities. Let's add (or, better, COMBINE) a fourth dimension.

First, how many directions have we added? Well, the first time, we added an infinity; and the second, we added an infinity of infinities. This time, we'll add three infinities of infinities. OK, now how about rotational planes? Well, first, let's name this new dimension. We'll call it "t." (No, don't jump ahead of me here. We'll get to it in just a little bit. Just accept it and don't wonder why for the moment.) So now how many planes do we have? Well, we had x-y, x-z, and y-z; looks to me like we now have added x-t, y-t, and z-t. So we didn't just add ONE more plane of rotation, we added THREE. Oh, NOW we understand why we couldn't envision four dimensional space; we only expected to get ONE more rotation axis... OOPS! I thought we weren't going to talk about them any more!

NOW do you see why we don't want those axes of rotation, and why we have such confusion as a result of them? Isn't everything MUCH clearer when we think about rotation PLANES? We have no problems understanding them; we can't quite visualize them, being 3D entities ourselves, but at least we can see how things must be in 4D space. This is as close as you'll ever get to being able to visualize 4D. If I've done well, then you can ALMOST see it in your head, and you can see how to manipulate objects in it.

Now how does that help with our 2D spherical surface that is curved through the third dimension? Well, knowing that adding that third dimension adds TWO more planes of rotation instead of just ONE, we can see that just adding that third dimension adds ENOUGH to let us do that curve. The problem you had before in defining just exactly WHY was because you were thinking that adding one dimension only added one axis of rotation, you see; but now that you know that it adds TWO PLANES of rotation, you can see why you only need one more dimension than the brane (that's a technical term for a manifold curved through an extra dimension, in this case a 2D brane curved through a third dimension) has to curve in.

OK, now what are the effects on a 3D brane of being curved through a fourth dimension? You can ALMOST SEE it. Just imagine a sheet of rubber, with various weights in various places- and then imagine the sheet THICKENING, but remaining curved in various places just as it was before it was thickened. And (this is just a little harder, but it's the last piece) imagine that even though the top and bottom of the thickness of the sheet aren't defining the curve in the middle of the thickness, THAT CURVE IS STILL THERE. And that's what it means to curve a three-dimensional manifold, like our normal conception of space, through a fourth dimension.

And because we now know that combining that fourth dimension adds not just ONE "AXIS" of rotation, but actually adds THREE PLANES of rotation, we can see how it provides JUST ENOUGH to do that to the manifold in.

I hope you found this helpful. I certainly have, and I can only repay the debt I incurred learning it by passing it on. If you have questions, ask away.

 

[Soapy Sam]

Q- If adding a second axis gives an infinity of directions to move in (rather than two orthogonal directions and a number of variants expressible therein), why say that adding a third axis adds a mere two orthogonal rotational planes, rather than an infinity of them?

I sniff a whiff of demonic inconsistency. (Interesting idea though, ditching the rotation axes. I do like that.)

 

[Just thinking]

Interestingly, adding ONE dimension to ONE dimension produced ONE plane of rotation; but adding ONE MORE dimension to that two didn't just add ONE more plane of rotation; it added TWO more. Now we have three planes of rotation.

Aren't there an infinite number of planes of rotation in 3 dimensions?

Perhaps you mean ... "Now we have three planes of rotation normal to each other."

 

[Just thinking]

Also, as you started off with, you mentioned that in 1 dimension there is only 1 direction of movement. I would argue that there are 2 (back and forth), but that may just be semantics. And when another dimension is introduced a new type of motion is achieved -- rotation. Is there a new type of motion achieved when going to three? ... cyclonic? If so, what might there be when going to four?

I ask this because a two dimensional being would observe unusual differences when observing a simple 2D rotation compared to 3D cyclonic motion. The 2D rotation would always appear the same -- but a 3D cyclone seen by the 2D observer in his 2D universe would display different parts of the object (a cyclonic funnel) as it moved through his 2D plane. Are there any similar or analogous objects we can observe (in our 3D universe) showing or hinting at higher dimensions (4D)? And what type of motion would this now introduce?

 

[epepke]

I ask this because a two dimensional being would observe unusual differences when observing a simple 2D rotation compared to 3D cyclonic motion. The 2D rotation would always appear the same -- but a 3D cyclone seen by the 2D observer in his 2D universe would display different parts of the object (a cyclonic funnel) as it moved through his 2D plane. Are there any similar or analogous objects we can observe (in our 3D universe) showing or hinting at higher dimensions (4D)? And what type of motion would this now introduce?

There are, of course, Lorentz contraction and time dilation, which indicate time as a kind of dimension.

However, you're looking for a spatial dimension. so you're obviously not looking for time.

There are a number of mathematical features that are worth pointing out. The cross product is closed (if that's the word I want) in 3-D, but not 2-D. I mean that if you take any two 3-vectors in a 3-D space and take their cross product, you'll wind up with another 3-vector in 3-D space (or else 0). This is not true in 2-D space. If you take the cross product of two 2-vectors in 2-D, you get a vector that is at right angles to any possible vector in the 2-D space (again, or else 0).

Things get even more interesting when you introduce quaternions. Special relativity is usually done in a Minkowski space, which is a flat space with a signature (-,+,+,+), where the - is for time, and the three + are for spatial dimensions. It also works, however, for a flat space with a signature (+, -, -, -), and that maps directly onto a quaternion, which very naturally model spacetime as a quaternion space rather than a vector space and also form a decent consistent algebra over the space.

So a quaternion is a kind of number that neatly models three dimensions of space plus time. Are there numbers that model other numbers of dimensions of space plus time? Yes, there are. The number like a quaternion that models one spatial dimension plus time I'm sure you already know about--it's a complex number. Is there one that maps onto two spatial dimensions plus time? No, there isn't. Nor four, nor five, nor six. Seven, yes.

Now, I happen to like these numbers, which are sometimes called Hamiltonian or post-Hamiltonian numbers (which is confusing, because some other things are called Hamiltonian numbers, and also a lot of other things named after Hamilton, who was a busy guy). What I mean are real numbers, complex numbers, quaternions, octonions, sedenions, and I forget the name of the last one. This is because one can rigorously construct algebras with them using the same basic tools that people use to construct the real algebra. They're not all the same algebra, though--quaternions and up are not commutative, octonions and up are not associative, etc. This is not necessarily bad, as the lack of commutativity in the quaternions is what makes cross products work the way they do with the right-hand rule.

In contrast, just piling up dimensions and putting signatures on them seems pretty ad hoc to me. There's no scientific guarantee that all the laws of nature have to be expressable in beautiful mathematics, and I have a prejudice in favor of beautiful math because most of my training was in math. However, in the past, there have been a lot of instances where following the math has led to good scientific hypotheses. So I'd like to see string theorists use post-Hamiltonian spaces rather than vector spaces, in the hopes that it can get rid of some of the unnervingly seemingly arbitrary aspects of string theory.

 

[Soapy Sam]

Octonions? For alphanumeric soup?

Epepke, after reading the above, I feel like lying down in a dark room with a damp towel over my head and having a little cry to myself.

My head is as big as anyone else's. Why is there no space in there for mathematics?