How Do Dimensions and Euclidean Geometry Relate to Spheres in Higher Dimensions?

[mewmew]

Ok, this is a really stupid question but it is something that is really confusing me. I am confused on the concepts of dimensions, for example of a sphere and how they relate to Euclidean geometry. A sphere in 2 dimensions using Euclidean 3 dimensional geometry doesn't really make too much sense to me. I always thought of a sphere as having 3 dimensions so I am confused. I understand if its written on paper it is obviously in 2 dimensions but I am still confused, and what about even higher dimensions, how do those work as far as Euclidean geometry? I'm just really confused and any help would be great. Thanks

 

[dextercioby]

For a differentiable manifold,we can define its dimension as the dimension of the tangent linear space in one arbitrary point of the manifold.

For the 2-Sphere,it's obvious,i think.There's also the theorem of embedding which asserts that any "n" dimensional diff.manifold can be immersed into $\mathbb{R}^{n+1}$.

Daniel.

 

[pervect]

Ok, this is a really stupid question but it is something that is really confusing me. I am confused on the concepts of dimensions, for example of a sphere and how they relate to Euclidean geometry. A sphere in 2 dimensions using Euclidean 3 dimensional geometry doesn't really make too much sense to me. I always thought of a sphere as having 3 dimensions so I am confused. I understand if its written on paper it is obviously in 2 dimensions but I am still confused, and what about even higher dimensions, how do those work as far as Euclidean geometry? I'm just really confused and any help would be great. Thanks

Does it make sense to you that if you have a 3-dimensional cube, it has a number of "surfaces" (six to be precise) that are two-dimensional rectangles?

 

[ohwilleke]

While the OP sounds a little garbled, I suspect what is going on is that some is saying that a sphere in three dimensions is analogous to a circle in two dimensions and that a person confined to a two dimensional surface might perceive a sphere as a circle. This analogy would be used to explain how a sphere in three dimensions in turn, might be perceived when in fact there is a four dimensional object present.

 

[mewmew]

Does it make sense to you that if you have a 3-dimensional cube, it has a number of "surfaces" (six to be precise) that are two-dimensional rectangles?

Let me say this for starters, so far I have gone through a years worth of college calculus so I know nothing of higher math yet. I haven't delt with anything like this before and am reading Hartles book on GR for some summer research.

So basically the surface is what determines how many dimensions it is? So a circle would be a 1-sphere because its surface is essentially a line, and a "sphere" would be a 2-sphere because its surface is like a plane? I guess I am just stupid because when I think of a sphere I think of it in x,y,z dimensions, a 3 dimensional object, but infact is just a 2 dimensional object, a plane for example, in a 3 dimensional geometry?

Edit: So if what I said above is correct, would a golfball be a manifold in n=2 because it can obviously be done in n+1 ,3, dimensional geometry? I guess for some reason this is harder for me to understand than a sphere(assuming once again what I said above is correct) because the surface isn't smooth, but dimpled, such it seems the surface to be in 3 dimensions? I don't know why I don't understand this stuff, it just confuses the hell out of me. Thanks for the help so far though.

 

[pervect]

Let me say this for starters, so far I have gone through a years worth of college calculus so I know nothing of higher math yet. I haven't delt with anything like this before and am reading Hartles book on GR for some summer research.

So basically the surface is what determines how many dimensions it is? So a circle would be a 1-sphere because its surface is essentially a line, and a "sphere" would be a 2-sphere because its surface is like a plane?

Yes

I guess I am just stupid because when I think of a sphere I think of it in x,y,z dimensions, a 3 dimensional object, but infact is just a 2 dimensional object, a plane for example, in a 3 dimensional geometry?

A sphere is a three dimensional object - but it's surface is a two dimensional object.

Since you've had collegee calculus, I assume you've seen where the volume of a sphere is integrated by taking intergal of the area of the surface times the radius

$$V = \int 4 \pi r^2 dr$$

Hopefully this intergal helps to convince you that the surface of a sphere is a 2-d object, when you multiple the surface area * depth, you get the volume of a small hollow sphere of thikness dr.

Edit: So if what I said above is correct, would a golfball be a manifold in n=2 because it can obviously be done in n+1 ,3, dimensional geometry?

Yes. As dexter already pointed out, all you really need to ask yourself is the dimensionality of a tangent surface.

If you have a sphere, the tangent to the sphere is a plane. The same is true for a golfball, or an apple. Any 3-dimensional object will have a 2-dimensional surface, as a matter of fact - with one small technical exception. If you have a sharp point. like a cone, the point of the cone would be a singular point, and not on the manifold. But if you restrict yourself to 3-d objects without sharp points, you'll be OK in saying that it's surface is a 2-d manifold.

 

[George Jones]

Ok, this is a really stupid question but it is something that is really confusing me. I am confused on the concepts of dimensions, for example of a sphere and how they relate to Euclidean geometry. A sphere in 2 dimensions using Euclidean 3 dimensional geometry doesn't really make too much sense to me. I always thought of a sphere as having 3 dimensions so I am confused. I understand if its written on paper it is obviously in 2 dimensions but I am still confused, and what about even higher dimensions, how do those work as far as Euclidean geometry? I'm just really confused and any help would be great. Thanks

I think it's just a matter of definition. Mathematic defines a sphere to the surface of what is often taken to be a sphere in everyday terminology. The (n-1)-dimensional sphere S^(n-1) with unit radius, which, when thought of as living in an n-dimensional Euclidean space, is defined as

S^(n-1) ={(x1, x2, ... , xn} | x1^2 + x2^2 + ... + xn^2 = 1}.

So, for a mathematician, S^2 is something like the surface of a beachball. Mathematicians do make sure there is no confusion between S^2, say, and

B ={(x1, x2, ... , xn} | x1^2 + x2^2 + ... + xn^2 <= 1}.

These are very different objects, and so are given different names. B and its higher-dimensional cousins are called balls. So, to a mathematician, a sphere and a ball are quite different objects, while in everyday life they might be taken to be synonymous terms.

The "volume" of the sphere S^(n-1) is (2 pi^(n/2))/gamma(n/2). This gives the "volume" of S^2 to be the standard surface area expression 4 pi.

Now, an aside. All spheres are closed and bounded, and thus, by Heine-Borel, compact. It is easy to show that S^3 and the Lie group SU(2) are equivalent (homeomorphic) as topological spaces, so SU(2) is compact. This means that its unitary representations (important in quantum theory) are finite-dimensional.

Regards,
George