intersection

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intersection

Postby Seldon » Fri Mar 23, 2007 4:59 pm

I wonder...is there an equation or theorem for the intersection of n-dimensional objects? The only thing I can think of is this:

An infinite figure of n dimensions that intersects another infinite figure of the same number of dimensions has an area of intersection in n-x dimesnions, where x is the number of dimensions that this whole thing takes place in.

However, this does not come close to encompassing all of the possible intersections, especially not in higher dimensions. Any ideas?

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Postby bo198214 » Sat Mar 24, 2007 11:00 am

I am not completely sure what you mean.
If we intersect two n-dimensional objects the cut/intersection can be of any dimension. What do you mean with "the whole thing"?
What we can say is that if we intersect an m-dim and an n-dim object the resulting dimension is at most min(m,n).

However in the theory of fractal and negative dimensions, they work with the probability of dimensions. Consider N-dim space with m-dim object A and n-dim object B then most probably is codim(AnB)=codim(A)+codim(B) where codim(X)=N-dim(X).

For example take two planes in 3d space the most probable cut is a line.
The codim of a plane is 3-2=1 and so the codim of the cut shall be 1+1 and indeed is this the codim of a line. Or cut of a plane with a line is most probably a point and indeed 3-2 + 3-1 = 3-0. Or a last example: the cut of a plane with a point should most probably be ... empty, but what dimension has the empty set? Lets compute: 3-2 + 3-0 = 3 - x. Hence x=-1 and this is where negative dimensions come into play.

Unfortunately in the moment I havent found a thourough introduction to this topic, but maybe have a look at http://classes.yale.edu/Fractals/ especially the page about dimension algebra. I think Manelbrot has introduced those probability and negative dimensions, at least he also has published about it.
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Postby Seldon » Sat Mar 24, 2007 3:52 pm

Let me rephrase. I am wondering if there is a formula or theorem that would state that if an intersection between 2 n-dim objects has dimension x, then this must occur in dimension y. (Sorry, I phrased it wrong before.)

However, the problem with this is that 2 planes can intersect in a line or in a plane in 3d, even though they are technically NOT in 3d when intersecting in a plane. Basically, I was looking for some kind of formula that accounts for that. (If that makes any sense.)
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Postby bo198214 » Sat Mar 24, 2007 11:12 pm

Seldon wrote:Let me rephrase. I am wondering if there is a formula or theorem that would state that if an intersection between 2 n-dim objects has dimension x, then this must occur in dimension y. (Sorry, I phrased it wrong before.)

However, the problem with this is that 2 planes can intersect in a line or in a plane in 3d, even though they are technically NOT in 3d when intersecting in a plane. Basically, I was looking for some kind of formula that accounts for that. (If that makes any sense.)


I still dont understand what you mean. 2 n-dim objects, regardless what dimension the surrounding space has (of course must be at least n), can have an intersection of dimension 0 till n.

What do you mean by the planes are technically not in 3d? Do you mean the sum of two planes is 2d if they intersect in 2d, while the sum is 3d if they intersect in 1d (where the "sum" is the set of all linear combinations of points in both objects)? So if you talk about "in dimension y" you mean y is the smallest dimension such that the union is contained in a space of that dimension? (instead of, as I had understood it, they are contained in a space of dimension y)

But then there is a counterexample consider two planes intersecting in a line and consider two parallel non-identical planes. In both cases your dimension y is 3 but the intersection has dimension 1 in the first case and dimension -1 in the second case.
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Postby papernuke » Sun Mar 25, 2007 1:03 am

I think i understand what you mean, but how would scientists make equation(s) out of dimensions that they cant observe directly?
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Postby Seldon » Sun Mar 25, 2007 1:45 pm

You got it right, bo198214. Its that counterexample that throws me off. The problem with what I had origionally said is that it is completely untrue if the figures are parallel. (What I was looking for was a way to compute the what dim two figures must be in to intersect in a certain way.)

As for not being able to create formulas/theorems for higher dimensions, that is obviously untrue. We all know that a tetracube has 8 sides, but we can't directly observe it. Same for lots of other stuff.
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Postby bo198214 » Sun Mar 25, 2007 3:10 pm

I didnt think about the case yet that all planes/lines/subspaces has to go through the origin. Maybe for this scenario one could indeed find some such laws ...
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Postby Seldon » Mon Mar 26, 2007 12:19 pm

How would that make it easier?
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Postby bo198214 » Mon Mar 26, 2007 12:46 pm

Then two parallel planes are automatically identical ...
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Postby Seldon » Wed Mar 28, 2007 6:13 pm

Oh, I see. Yes, I suppose it is possible to create formulas for that scenario.

The only thing I can think of is a simple little equation that pretty much says what we already know:

n + x = y

n=dim of objects
x=dim of area of intersection
y=dims required for intersection (i.e., 2 planes intersecting in a line require 3 dims)
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