Visualizing intersecting multidimensional objects.

[Keys]

If we look at 2 intersecting orthogonal planes in 3D, the intersection forms a line if you are "living" on either plane. How would the intersection look if there are 2D planes in 4D where the planes do not share a dimension? For example plane 1 exists on X and Y, and plane 2 exists on Z and T. I'm figuring it must be a point if viewed from either plane. Is this correct? The other answer might be there is no intersection, but I don't think this can be correct.

 

[mathman]

This belongs in the math forum.

One way to approach it is to consider the representation of the plane in parametric form, such as: X=A+Bu+Cv, where u and v are parameters, while A, B, C are constant vectors and X is a point on the plane.

In 3 space, when you have 2 planes, you will have 3 equations in 4 unknowns (the (u,v) parameters for each plane). The solution is then a line. In 4 space, you have 4 equations in the same 4 unknowns, leading to a single point.

 

[Keys]

Thanks Mathman for your answer. I'm a bit dense, does this parametric form work even higher dimensions? Let's say the last example 2 planes in a 4 space, could be considered 2 different 2 spaces both within a 4 space. By my reasoning a 3 space and 2 space that are both within a 5 space where they don't share any common dimensions, the intersection would appear as a line in both the smaller spaces in question. With two 3 spaces within a 6 space which don't share common dimensions, the intersection in each 3 space would be a plane or a common 2 space (which is not comprised of any of the discrete dimensions which make up the 6 space). Thanks

 

[mathman]

The parametric form works in any number of dimensions. One parameter gives a curve (1 dim.), two parameters a surface (2 dim.), three parameters a solid (3 dim.), etc., where the number of dimensions of the underlying space is the same as the dimension of the vectors.

To get "flat things", the parameters appear as first powers only. Other objects will have higher powers, functions (such as sin, cos), and functions involving products of parameters, etc.