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