As I played around with 4D toys, I did start to gain a sense of things that were "Behind" or "In Front" of me in 4D space, that is I could start to feel their p

As I played around with 4D toys, I did start to gain a sense of things that were "Behind" or "In Front" of me in 4D space, that is I could start to feel their presence after they had been pushed out of sight, if I knew in what direction they were moving in the 4th dimension. If I didn't know that I could only feel that they were near. It helps that there is a little visualization that is just a line-per-object showing where you are, and where all the objects are intersecting the 4th dimension, that you also use to move "back and forth." One thing I found myself doing was grabbing objects at one of their edges in the 4th dimension, by moving myself to near their boundary, and then using them like brooms. It's really easy to understand with the case of a hypersphere, since at its edge it's just a smaller sphere than at the middle. So you grab that small sphere at the edge, and push in the direction towards its middle in the 4th dimension, and it will act like a bowling ball. You won't see the stuff you are pushing around because the sphere is "ahead" of you, unless they roll around the sphere, then you'll pass them. Once you reach the edge of the 4th dimension, all the stuff you kept pushing will be there. Predicting how 3D intersections change as objects rotate about in the 4th dimension still seems like chaos to me though, except in the case of hyperspheres, which basically don't change as they rotate, but I only played around for about an hour or so. The only way I found to rotate objects in the 4th dimension was to have them collide with each other, or the walls and floor, which makes it kind of hard to carefully experiment with their rotations.