How is a 2-sphere in a 3 dimensional space?

[fk378]

How is a 2-sphere in a 3 dimensional space?
I do not understand how, according to wikipedia, a 2-sphere is a "2-dimensional surface (which is embedded in 3-dimensional space)."

Why is it not a 3-dimensional surface, since we need 3 coordinates to determine a point on the sphere?

 

[latentcorpse]

no. you only need two coordinates. r is fixed therefore any point is given by the coordinates $\theta,\phi$. this generalises to hgiher dimensions.

i.e. $S^{n}$ is embedded in n+1 dimensional space.

 

[fk378]

In class we had an example where U is the set of all vectors x with n+1 coordinates in the n-sphere. How can there be n+1 coordinates in an n-dimensional sphere?

 

[HallsofIvy]

In class we had an example where U is the set of all vectors x with n+1 coordinates in the n-sphere. How can there be n+1 coordinates in an n-dimensional sphere?

One of the coordinates must be a function of the other n. In the example of the "two sphere", we can identify all points as $(\rho, \theta, \phi)$ using spherical coordinates. But $\rho$ is a constant, the radius of the sphere.

Another example is the plane through (1, 0, 0), (0, 1, 0), and (0, 0, 1). The equation of that plane is, of course, x+ y+ z= 1. Any point on that plane can be labeled (x, y, z) but we can write anyone of those coordinates in terms of the other two. For example, (x, y, 1-x-y). Another possiblilty would be (x, 1- x- z, z). Three coordinates, but written in terms of two parameters- a two dimensional surface imbedded in a three dimensional space.

 

[fk378]

So if you write it out as (x,y,z) is it still considered to be a 2-dimensional surface? The point is that you *can* write it out in terms of 2 parameters, is this correct?

 

[VeeEight]

Yes this is correct, as stated in HallsofIvy's last sentence.