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<br> &lt;a,b | [a,b] is central &gt; 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!

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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...

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Here's a model I once made:
[![Contact structure][1]][1]<br>
and here's the same model upside down, before it was completely finished:
[![Contact structure][2]][2]


  [1]: https://i.sstatic.net/sAPYS.jpg
  [2]: https://i.sstatic.net/rdVqA.jpg