The first answer is yes, but my experience all relates to sculpture and mathematics, deeply related but not very practical.
The best way to start is to remember that every spatial relation you can dream up and or construct as a space, also has an associated logic. The technical method to demonstrate this equivalence is through group theoretical methods of morphisms and representations.
You can see this directly with the isomorphism that is establish between logic and Venn diagrams. For each combination of boolean operators and sets there is a corresponding Venn representation. The two circle overlap is the diagram for the boolean set operator of intersection. The three circle diagram is i /\ j /\ k, where i, j, k are circles (point loci sets) and /\ is the intersection operator.
The move from 2-d to 3-d changes the topological and dimensional characteristics of the space. You move from rounded triangles to convex tetrahedrons.
With that general math machinery in mind the question becomes what kind of relationship and its logic do to you want to represent in 3-d (or the illusion of 3-d) that isn't available in 2-d? Or is this all about the prettiness of 3-d graphics?
Side bar. What is represented beyond the circle? Here is how to imagine the answer. Put a center dot in the circle. You have just added the point at infinity. It is the point at infinity, which is the mirror of all the points at an infinite distance from the boundary outside the circle, under the conformal transformation of the imaginary plane. Every line and point outside the circle can be represented by a point or line (or shape) inside the circle.
The way to physically construct this system uses elementary methods of plane geometry, with straight edge and compass, following the theorem on constructing a polar to a point with respect to a circle. You can also do it with elementary algebraic geometry---the equations are very tedious, but the essential concept is the same, i.e a conformal mapping.
Here's the cheap and practical bottom line. You want to figure all the concrete theory with a pencil and paper and make a sketch. Then simply use Adobe Illustrator to reproduce the geometric idea in a nice clean diagram. Then you want to put a box under the illustration and explain what the illustration means.
Now skip the concept of a Venn diagram and think more simply. All you want is to take the intersection of three circles in Illustrator and turn them into an intersection of three spheres. All the work has already been done and if you have photoshop and illustrator, you already have the tools.
Go here to blow your mind. An old hippy art major wakes up from his dream of being a philosopher of action...
http://en.wikipedia.org/wiki/CMYK
Now type in google, ``rbg color model, 3-d'' and look around for ideas.
The point is you can use a combination of color models in your graphics program together with a vector graphics program to make your illustration, or rather the 3-d representation of your logical model.
If you want the animated version, you have to upgrade to the animated video software. And your mathematical tools have to upgrade to analytic functions, Fourier transforms with the physics of thermodynamics. How bad do you want it?
This kind of stuff is very sophisticated in physics, chemistry and statistical mechanics. Think 2-d color representations of 3-d phase spaces and the like. These richly endowed institutions can take statistical data and generate a picture in color of what is going on in the molecular and atomic levels. I discovered all this on 9.00/hr. So, ehem, let's don't hear any shit about being too serious.
Just checked Sandy Harris's link. Yeah, like that ...
CA