I've played with some programs I made to explore 4D objects, and, yes, you do gain some intuition. Interestingly, I was told one of George Boole's daughters go

I've played with some programs I made to explore 4D objects, and, yes, you do gain some intuition. Interestingly, I was told one of George Boole's daughters got lessons from a guy on 4D objects by using colored wooden blocks. She really picked it up, and later in life mathematicians would consult her about intersections of 4D polytopes.