tl;dr What 3D shapes can you get by slicing a hypercube? What 2D shapes?
I fell into a deep hole of hypergeometry after talking to some high schoolers about hypercubes (specifically a 4-cube, a tesseract). I wrote my own mid-level explanation of some of the weirdness of such a hypercube trying to explain how one of the more common 3D projections of the hypercube actually signify, and ended up with a ton more questions that I started with, some of which I’m figuring out the answers to myself, but some that I’m even failing to google answers for.
If you look at the intersection of a 2D plane and a 3D cube you can get triangles, squares, pentagons and hexagons, depending on the angle and position of the plane.
What do we get as the intersection of a 3D hyperplane and a 4D hypercube? I can figure out that if let all the edges be parallel to an axis, and our hyperplane is perpendicular to one of those axes, the intersections will all be cubes. But the concept of having what is essentially a volume perpendicular to anything is mindboggling in and of itself, and I’m not finding a good source online showing what happens when you do the same at an angle.
And if a hyperplane isn’t a volume, then what is?
I’m also curious what happens if you look at the intersection with a 2D plane.
I’ll post the link to my hypercube explanation in a reply if anyone is interested, but this isn’t a promotion of my blog.
Not only that – if you slice different planes through a 3D collection of tightly-packed and stacked cubes, you can get a plane tessellated by regular hexagons, a plane tessellated by regular squares, or a plane tessellated by regular equilateral triangles. (You can also get tessellations by rectangles and, I think, non-equilateral triangles).
No slice through a cube will get you a regular pentagon, though – there aren’t any planes with five-fold symmetry in a cube. If you can get a pentagon, it’s not a regular one.
the Wikipedia page on the tesseract gives projections of the tesseract into 3-space. I realize that this isn’t necessarily the same as “slicing” a hypercube with a 3-space but it’s the closest thing I could find
Anything you can get by slicing a 3d cube with a plane, you can use as the base of a prism, and then get that prism by slicing a 4d cube with a 3d space. So you could also get (for instance) a triangular or hexagonal prism.
Though that doesn’t exhaust the possibilities, either.
