Hypercubes

I've been planning a "mirror matter" post for a while now but it's so fun to research that I haven't had the discipline to write about it. Hopefully I'll get around to it soon.

In the meantime, I've been reading "Surfing Through Hyperspace" by Clifford Pickover. It might be the strangest book I've ever read, but it does pose some very interesting questions. The book deals with understanding higher dimensions through a second-person narrative science fiction story interspersed with more technical explanations. There is an emphasis on 4-dimensional beings and what it would be like to interact with them. We would only see 3-dimensional blobs changing in size. But a 4-dimensional being looking at us would be able to see everything; they would be able to look at us inside and out all at once. A 4-dimensional being would be much like a god: able to see and hear everything at once; would be incapable of being seen; and it would have miraculous powers, like the ability to remove a tumor without making a single incision. I found it interesting that immediately after, Pickover pointed out that a 4-d being could also impregnate a virgin.

Later on, he discusses hypercubes: 4-dimensional cubes. I've seen pictures of 4-dimensional cubes projected into 2-dimensions. I used to draw them all the time when I was bored in class.
But he also gives beautiful images of 5-d, 6-d, 7-d, and 8-d cubes projected into two dimensions, that I've actually never seen before.

A 5-dimensional cube:
A 6-dimensional cube:
A 7-dimensional cube:
and, an 8-d cube:
What I found even more interesting was a graph he gave of the volume of a radius 2 sphere as a function of dimension.
The x-axis corresponds to the number of dimensions of the sphere, and the y-axis, to the volume of the sphere. The peaks occur at different places depending on the radius of the sphere.