I always thought of the 4th dimension as time. So i guess that would be animation.
Slightly off-topic, but a random concept if anyone is interested: (Not necessarily graphics specific)
Maybe you could introduce 4th, 5th, and 6th dimensions that are representations of some form of energy. So just like something can grow taller, wider, or gain depth; it can gain more of Energy Y or Energy X, or Energy Z.
So an object from the X/Y/Z 3D perspective might look like a sphere, but maybe this object from the perspective of the EX/EY/EZ 3d perspective might look human or like an animal.
(I guess this could also used to represent spirituality, also)
Game-wise: You have to get the ball into one of the holes. But for some reason you cannot get it into some. Maybe when you press some button, you can switch from X/Y/Z into EX/EY/EZ, and you see a different world and different overall form of the object.
Edit: So maybe a 4-6D graphics could be reresented by an entirely separate 3d model. (Trying to relate with the original post somewhat?)