Notices
Results 1 to 5 of 5

Thread: Dimensional Superposition

  1. #1 Dimensional Superposition 
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    I think we sorely need an operation I call "Dimensional Superposition", symbol "&". i.e. to construct n dimensional spaces from n-k dimensional spaces.

    For example we may use it to construct a 3D space using two 2D spaces (in flat space) by defining two orthogonal veftor in each plane. Then do DS by superimposing two of them from diffetent planes such that the other two is orthogonal and the origin of the two pairs superimpse.

    This increases the tools and may generalise to curved spaces.

    S = (uv)&(vw).

    It would also enable derivation of properties in higher dimensional spaces that have no analog in lower dimensional spaces.


    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  2.  
     

  3. #2  
    Forum Junior c186282's Avatar
    Join Date
    Dec 2008
    Posts
    208
    Good idea! but sadly like most good ideas someone has done it:
    Take a look at:
    Exterior product


    Reply With Quote  
     

  4. #3  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    The wedge product increases the grade of a vector. I don't see any construction of higher dimensional space from lower dimensional one(s) at that reference.
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

  5. #4  
    Moderator Moderator
    Join Date
    Jun 2005
    Posts
    1,614
    Yeah, the wedge is irrelevant here, but we do have such an operation on an inner product space.

    Suppose that is a vector space with n basis vectors. Define the space as the vector space spanned by the k bases chosen together from n (where k < n)

    Whenever is an inner product space we may have the so-called Hodge operator .

    This n - k space is dual to the original space, so there is an isomorphism , and so that as a bijection i.e. . (Note this is an edit)

    Just watch out for signs......

    Any use?

    Edit: I didn't explain that very well, even though I'm not sure this is what talanum is after. Anyhoo, more detail here
    Reply With Quote  
     

  6. #5  
    Forum Sophomore
    Join Date
    Jul 2007
    Location
    South Africa
    Posts
    196
    Hodge star does not do it. I construct grade 1, n dimensional spaces from grade 1 n-1 dimensional spaces. See:

    http://www.scribd.com/doc/21065610/4...uct-Derivation
    It also matters what isn't there - Tao Te Ching interpreted.
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •