Thread: Dimensional Superposition

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.

Good idea! but sadly like most good ideas someone has done it:
Take a look at:
Exterior product

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.

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

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