My last post was about the different ways that two lines can relate to each other in Euclidean space. Unfortunately the main point I was trying to make was entirely false, so I’m going to explain why.
The error began when I said that it should be possible to position two lines in 4-dimensional space such that they cannot fit into any 3-dimensional subspace. That’s not true, and so everything I said after that is also false, and the entire post isn’t all that interesting anymore.
The following brief explanation should do fine for anybody with a good understanding of linear algebra:
One line can be defined by naming any two distinct points on the line, and so we can define any two lines by naming two distinct points on each of them. I’m not going to bother distinguishing which two points belong to which line, because I can generally show that given any four points, any two lines passing through two points each will necessarily fit into a 3-dimensional subspace.
Given any four points in N-dimensional space, translate the four points together so that one of them is positioned at the origin. Now we have three points that are not at the origin, which we can consider to be vectors. Define a vector subspace using these three vectors as the basis. Clearly, with a three vector basis, the vector subspace cannot have a dimension higher than three. Now if we have a 3-dimensional space that contains all four points, it must be the case that any line through any two of those points must also be contained in the space (this can be shown more strictly by stating this in terms of basic vector addition and subtraction).
So far we’ve defined a vector subspace for our four translated points (we translated them because a vector subspace must contain the origin). To apply the result to the four original points, we can simply translate the entire subspace using the reverse of the original translation. The resulting space will not be a vector space, since it will not contain the origin (unless of course one of the four points happened to be the origin), but it will nonetheless be a 3-dimensional space, containing the original two lines.
For those without a linear algebra background, this ought to be convincing enough, though not as mathematically rigorous:
No matter the dimension we’re talking about, if we pick any two distinct points, there should be only one line through the two points. So we can define each of our lines with two points, giving us a total of four points. In the same way that two points define a line, three points can define a plane (assuming they aren’t all in a straight line). And in the same way that three points define a plane, four points that don’t already lie on a flat plane will define a specific 3-dimensional space. This isn’t immediately obvious, because we only have three dimensions, so any four points we can imagine can at best define the 3-dimensional space which is the whole of space as we know it. But in any higher dimension this will be more meaningful. So our four points from our two lines define a 3-dimensional space containing them, and if some 3-dimensional space contains the four points, it will also contain any line through any two of them.
It’s not a very difficult thing to see, but I didn’t catch myself until after I had posted it. I need to be more careful with my maths.