Volume of a 4 dimensional orb equation

[BlackJack]

In a lecture today, we (my professor) calculated the volume of a 4 dimensional orb. I discussed it with a few other people but no one could properly explain how to picture this orb.

What do you think does it "look" like?

One possible theory: An orb which depends on the time and shiftes it's shape accordingly. But this doesn't seem satisfactory to me.


Help ?

 

[Moo Of Doom]

You CAN'T imagine what a 4D orb "looks" like. I can think of no PhD who has been working with higher dimensions for a long time who would admit to being able to picture a 4D object. The closest thing you can get, pictorally, is to slice the object into 3D slices and associate each slice with a different 4th coordinate. In that case, a 4D orb starts at a point at (0,0,0,-r) grows in a circular fashion as you move towards 0 along the 4th axis until it forms the sphere x^2+y^2+z^2=r^2 centered at (0,0,0,0) and then shrinks again in a circular fashion until you hit (0,0,0,r). The radius of the 3D spherical slice at the 4th coordinate (let's say w) w=a where -r<a<r is going to be sqrt(r^2-a^2). That's pretty much what a 4-sphere is.