# What's the actual shape of the upper dimensions?

1. Feb 16, 2016

### Amine_prince

first i would like to ask a question .. imagine there are 2 dimensional creatures that are able to live and interact . they can exist in the second dimension with no problems . can the surface of a sphere precieved as a valid 2 dimentional space that these creatures can exist in ?( if yes then continue reading , if no then i am making an invalid point here). the problem with that space is the fact that it's bent in the third dimension . if this 2 dimensional creature would go in any direction in a straight path , he will return to his main position over and over again .

now , for this to be applicable to the 3 dimensional space , it has to be bent in the 4th dimension .
if the surface of a sphere is the 2 dimensional world existing within the 3rd dimension then the infinite number of Surfaces of the infinite number of 3 dimensional spheres should form the 3 dimensional surface of the 4 dimensional sphere . so the 3rd dimensional space that can have such properties is the surface of the 4 dimensional sphere ... then if i would be living in the bent 3 dimensional space and i move towards any direction after reaching the edge of the universe (i wouldn't even notice that an edge exists) i would return to my starting point all this is caused by the space is being bent in 4 dimensional space .

now how can we actually figure that space isn't in someway bent in 4 dimensional space ?

2. Feb 16, 2016

### MrAnchovy

With you so far...
I don't understand any of this, I'll just assume you are talking about a 3-sphere.
That is one of the most important unresolved questions in physics.

3. Feb 16, 2016

### jbriggs444

It is, perhaps, easier to think about 3-space being bent through a fourth dimension. But that fourth dimension is optional. One can describe curved spaces without requiring that they be embedded in higher dimensional spaces.

I had a very tough time parsing that. But yes, it is possible to have a three dimensional space which folds back onto itself so that it is both finite and without boundaries.

There is no way from within a space to know whether or not there is a higher dimensional space within which it is embedded.

4. Feb 16, 2016

### Amine_prince

yes , but you need an other dimension , it's impossible for all that to be done within dimension 3 .
a 4 dimensional sphere is an infinite number of 3 dimensional spheres with a variation of radius along the line

5. Feb 16, 2016

### Khashishi

From what I can gather, most present-day physicists have taken the philosophy of, if you can't measure it, even in principle, then it isn't real. Physicists do sometimes include unmeasureable things in their theories, but they refer to these aspects as interpretations or pictures and not truth.

There might be additional space dimensions out there responsible for the curvature of space-time, but we don't know of any way to measure them, so for now they are only considered to be part of a mathematical model (to help us understand things) and not really there. Now, if someone comes up with a way to measure these things (even in principle), then things will change.

6. Feb 16, 2016

### Amine_prince

basically , even if this is in some sort real , it's impossible for a 3 dimensional entity like us to measure it or figure it in any sort , but it remains interesting .
and when speaking about a 3 dimensional space being bent in a superior dimensions , when you get time and motion into it , it will all turn into a large mess with no solutions .

7. Feb 16, 2016

### mathman

The 4th dimension is unnecessary. In your example, the surface of a sphere is 2 dimensional, but it does not require a 3rd dimension to be defined.

8. Feb 16, 2016

### SardonicSparrow

I think the current outlook is unreasonably pessimistic. Just because the solution hasn't been conceived of, doesn't mean it's inconceivable. Additionally, when conceptualizing the fourth dimension you should be careful not to forget that it isn't a spatial dimension, and thus doesn't follow the rules of spatial dimensions. As far as moving four-dimensionally goes, you are doing it at this very moment, moving at approximately one earth-second per second. The problem with accurately conceptualizing time is that A, time is relative based on your proximity to gravitational fields (ie Time Dilation), and B, it is the only plane in which everything seems to be moving unilaterally in one direction.

9. Feb 16, 2016

### Staff: Mentor

The subject of Riemannian geometry is the study of the intrinsic geometry in curved spaces without reference to any external Euclidean space. So it is not necessary to assume the existence of the third dimension at all in the description of the geometry on a sphere.

A flat higher dimensional space is called an embedding space.

10. Feb 16, 2016

### Staff: Mentor

This is simply not true. General relativity has many solutions using the math of Riemannian geometry. Embedding spaces are not required.

11. Feb 16, 2016

### Amine_prince

well ... can you guys give me a 3 dimensional model .. in which if i go in any direction in a straight path i end up at the starting point ?
can you do that without using a 4th dimension ? i don't think so .

12. Feb 16, 2016

### Staff: Mentor

Sure. A 3 torus and a 3 sphere are two common examples. The three torus can even be flat.

EDIT: also, the property you are asking about is more of a topological property than a curvature property

Last edited: Feb 16, 2016
13. Feb 16, 2016

### Khashishi

@Amine_prince The problem is that you've assumed that space is Euclidean, so a non-Euclidean geometry is impossible from the start. Nobody can go out and 3D-print or carve a non-Euclidean 3D shape and hand it to you, but that doesn't mean the universe itself isn't non-Euclidean (oops, is that a triple negative?). Yes, it is possible to visualize curved 3D space as a surface in a flat 4D space, but unless we can figure out how to move or look outside the surface, then we shouldn't assume it actually exists. (If you want to include relativity, go ahead and replace 3D with 4D, and 4D with 5D, and Euclidean with Minkowskian.)

14. Feb 17, 2016

### Amine_prince

the movement is in the third dimension , i can also move up , no object that can exist within a single 3 dimensional space can verify that property . it's impossible .

Last edited: Feb 17, 2016
15. Feb 17, 2016

### Amine_prince

the problem with Bent space , is the fact that it's constructed within the 4rth dimension , let me take some time to show this .
now l'ets take the shape in any dimension in which any point of it has the same distance from the center . in dimension 2 it's a circle and the circle is very interesting because the uni-dimensional entity can only move in one path , that's supposed to be a straight one , but if you bend it in dimension 2 and make it a circle that entity will always return to the starting point . now the 2 dimensional entity can flow in 2 directions instead of one . for you to give it a space that would verifty the main condition the space has to bent 3 dimensionally , so that when it flows in any direction in any sort it will always get back to the starting point , so here space being 2 dimensional is actually an illusion , that's because the movement of the 2 dimensional entity has an equivalent movement in the next dimension , the the 2 dimensional entity is simple incapable to control or even notice that .so the 2 dimensional entity itself is actually in some way bent . it's not actually a 2 dimensional entity at all , it's like a flat 3 dimensional creature that has no control of it's movement in the 3rd dimension .

now , if this would be applied to us , we would be flat 4 dimesnional creatures , and we would have no control over our movement in the 4rth dimension , and the one and only shape that can allow the property to exist is a 4 dimensional sphere .

that's just a particular case , in which space is actually bent in a specific way to get the illusive 3 dimensional entity (that is actually a flat 4 dimensional entity in this case) to return to it's main position every time .

now if we suppose that we actually are flat 4 dimensional creatures , then the illusive space would be able to bent in other odd different ways that will add some even odder properties to the flow of flat 4 dimensional creatures in space .

if i read this again i might not understand it .

16. Feb 17, 2016

### Staff: Mentor

Yes, you asked for a 3D space, so of course the space is 3D. However, neither a 3 sphere nor a 3 torus require a 4th dimension. For example, the metric on a unit three sphere can be written:
$ds^2=d\psi^2 + \sin^2 \psi(d\theta^2+ sin^2 \theta \, d\phi^2)$
Where the coordinates
$0\le \psi < \pi$
$0\le \theta < \pi$
$0\le \phi < 2\pi$
Show both the "wrap around" you specified as well as completely specifying the geometry in the 3 sphere without any reference to a 4D or higher embedding space.

17. Feb 17, 2016

### jbriggs444

Dale beat me to the punch. This version is aimed at a less sophisticated audience.

To the extent that post #15 is intelligible at all, it is a demonstration that a finite-but-unbounded n dimensional space can be constructed by embedding that space in an n+1 dimensional Euclidean space. It does nothing to demonstrate that this must be the case.

The mathematical discipline of "topology" deals with these kinds of questions. A topology on a set of objects (for instance the points on a circle) determines which points are "close" to which other points. One can define this notion of closeness with a "metric" -- a numeric measure of closeness that obeys certain rules such as the triangle inequality. [More generally, one can define the notion of closeness with a collection of "open balls" -- neighborhoods around each point that may or may not may not contain other points, but that is generally more difficult to wrap one's head around]

In the case of a circle, one can take the set of objects to be the set of points in two-space that are at radius r from some selected origin. And one can define the metric distance between two objects to be the [positive] angle formed between those two objects as measured from the origin. But one does not need to do this.

18. Feb 17, 2016

### weirdoguy

Of course you can, most of the differential geometry books show that, it's a basic fact.

19. Feb 17, 2016

### Khashishi

I have a feeling Amine won't be convinced by examples of 3-manifolds because these examples can also be embedded in a higher dimension. Frankly, there's no test to show if our universe lies in a higher embedding dimension. The only reason to say that there isn't an embedding dimension is Ockham's razor. If you assume the existence of another dimension, then you open up all these questions that we don't know how to answer. Like, what is 1 meter from us in the w direction? Are there other universes ("branes?") out there? Can we interact with things outside our space-time? Physics is about modeling reality, and these models have to be tested by experiment. The models should be simple, but they do not have to be intuitive. Can you think of an experiment which would test the existence of a higher dimension? We are biased toward Euclidean geometry, and perhaps you view Euclidean space as a Platonic ideal or perfect form. But doesn't really serve as a valid basis for physical models, because sometimes physics is not intuitive, and your idea of ideal is kind of arbitrary and might disagree with others.

20. Feb 17, 2016

### Amine_prince

thank you very much sir .