That 112-dimensional cutout is so unrealistic

UPDATE: The mathematical content of this post is, as far as I know, original to me, and I posted it shortly after it occurred to me, without thinking through it critically. I later realized that it is entirely false. My next post explains why.

I’ve always enjoyed imagining higher dimensional geometry. Not that it’s easy to visualize, but it’s easy to discern what some of the properties would be. Today I thought of an example of such a property.

Consider the different ways two (infinite) lines can be positioned relative to each other in different dimensions:

0. Lines can’t exist in 0-dimensional space
1. Any two lines in 1-dimensional space are necessarily the same line
2. In 2-dimensional space, two distinct lines can relate to each other in two different ways:
   • They can intersect
   • They can be parallel

3. In 3-dimensional space, there is a third option:
   • They can intersect
   • They can be parallel (which means they are always the same distance apart)
   • They can be skew (if this one isn’t immediately obvious, imagine a bridge, where one line (the road) goes over the other line (the river), and they neither intersect, nor run parallel)

   The first two options imply that the lines are co-planar: there is some flat (2D) plane containing both of them. In the third option the two lines are necessarily not co-planar.

4. If we add a 4^th dimension, we should expect to be able to distinguish between two different kinds of skew lines:
   • Skew lines that can be contained within the same 3-dimensional sub-space, in the same way that parallel lines can be contained within a 2-dimensional subspace. We can refer to them as Skew₃
   • Skew lines that cannot be contained within the same 3-dimensional sub-space, in the same way that Skew₃ lines cannot be contained within a 2-dimensional subspace. We can refer to them as Skew₄

   Intersecting and parallel lines will both still be possible, of course.

5. With the 5^th dimension too, we should be able to have Skew₅ lines, which cannot be contained in the same 4-dimensional subspace.

And up and up, as high as you like. Our fellow beings in some 307-dimensional world probably spend most of their high school geometry class memorizing the 307 different ways that two lines can relate to each other (my favorite is Skew₁₀₈), and developing their ability to quickly recognize them when projected in a 307-dimensional hologram.