I've once made a 3D model of a contact structure. This is a remarkable object, and the feeling one gets by looking at it is difficult to describe in words. I've already spent a lot of time pondering at the idea of making a big sculpture out of it (including going and talking with someone whose job is to make metallic constructions).
Here is a mathematical description the object: take the Cayley graph of the Heisenberg group
<a,b | [a,b] is central > and embed it in 3D. This Cayley graph is infinite, and I'm of course imagining taking a finite portion of it (5x5x5 nodes works pretty well). The vertices are 4-valent, and at each vertex, the directions of the four incident edges are coplanar. On those four edges, you then position a small piece of plane: that's the contact structure!
If you want to make this object interactive, you could imagine little cars moving on it, their x-y-coordinates could be somehow specified by the user...