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Homework Help: How is a 2-sphere in a 3 dimensional space?

  1. Jan 30, 2010 #1
    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?
     
  2. jcsd
  3. Jan 30, 2010 #2
    Re: 2-sphere

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

    i.e. [itex]S^{n}[/itex] is embedded in n+1 dimensional space.
     
  4. Jan 30, 2010 #3
    Re: 2-sphere

    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?
     
  5. Jan 31, 2010 #4

    HallsofIvy

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    Re: 2-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 [itex](\rho, \theta, \phi)[/itex] using spherical coordinates. But [itex]\rho[/itex] 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 any one 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.
     
  6. Jan 31, 2010 #5
    Re: 2-sphere

    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?
     
  7. Jan 31, 2010 #6
    Re: 2-sphere

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