The latest game I've been playing is "4-D Toys." Take a zero-dimensional dot and extend it horizontally to make a one-dimensional line; take the line and extend it vertically to make a two-dimensional square; take the square and pull it out of a page to make a three-dimensional cube; then take the cube and extend it into a fourth dimension (at right angles to the first three), and you get a hyper-cube (or a tesseract, or a 8-cell).
4-D Toys allows you to manipulate objects, like tesseracts, which extend into four dimensions. Where it gets confusing is that it's possible to bump objects out of the three-dimensional slice that's being displayed ... sometimes you can bump objects back into your particular three-dimensional slice with another 4-D object. Just like a square is a 2-D cross-section of a cube, a cube can be a cross-section of a tesseract; knowing that abstractly does nothing to mitigate the trippy feeling of watching a cube contract into a wedge, fall over, and apparently disappear. Don't get me started on hyper-spheres; I'm still not sure how those work, other than spheres can shrink or grow, sometimes in the apparent mid air.
I was dreaming about tesseracts the other night; no special meaning for them that I can think of, I think I was merely practicing moving them around.
After playing around with tesseracts and hyper-spheres, I wondered why we don't have objects that extend into the fourth dimension -- brane theory suggests that there are additional dimensions, so why don't we have hyper-keys that can fit into hyper-locks?... or maybe photons and other really small quanta are interacting with each other on some micro- hyper-dimension, and what we call forces are actually higher dimensional kinetic interactions... But no... if that were true I'd expect inter dimensional friction to unexpectedly slow things down and heat them up....
At this point, I'm feeling like it's time to re-watch Dr. Strange.