Visualizing the 4th dimension is fun to think about
, and if you take calculus 3 you start talking about this kind of stuff. Here is the equation of a 2d circle
x^2 + y^2 = r^2
x is the independent variable, y is the dependent, and r is the radius constant variable. Now for a 3d sphere we have the following equation
x^2 + y^2 + z^2 = r^2
Now x and y are both independent variables, and z is dependent on x and y, and r is still a constant variable. So a 3d coordinate system looks like this
(the cube in the center is for visual aid)
this coordinate system is composed of 3 planes, xy, xz, yz, so when you're graphing any old function like
f(x) = x^2 this function graph is sitting on the z = k plane where k is a constant.
So if z = 0 we get the 2d circle again. This is the level curve at z = 0. And the range of z will be -r <= z <= r, look at the case when z = r.
z^2 = r^2
x^2 + y^2 = 0
This is a 2d circle of a radius 0, which is nothing. Now playing around with a 4d equation...
x^2 + y^2 + z^2 + q^2 = r^2
where x,y, and z are all independent variables, and q depends on x, y, and z, and r is still the radius variable. This is an equation of a hyper sphere, and can't be graphed directly, but can be observed in a similar way we observed a 3d sphere.
if we let q = 0, we obtain the 3d equation x^2 + y^2 + z^2 = r^2, Instead of level curves we have level surfaces, where the 3d sphere is the surface at different levels of q = k where k is a constant. Look at it like this
x^2 + y^2 + z^2 = r^2 - q^2
so as q increases and approaches r the radius of the sphere will be getting smaller and smaller. So a hyper sphere can be thought of as a sphere within a sphere and so on like a 3d sphere can be thought of as a circle within a circle, etc.
Now if you wanted to find formulas such as the volume of a hyper-sphere you would have to do a triple integral of the hypersphere equation. This get's ugly, not so much the integration part, but finding the domains to integrate over. :S
If you like thinking about higher dimensions you would definitely have alot of fun in calc 3.