Jan '05 - Mar '06
A presentation by Michael Silver this afternoon on new fabrication technologies got me musing about the mathematical problems involved with the use of two computer-controlled machines he showed: a hot-wire foam cutter and a bandsaw.
With both machines, you're bound by the limits of only being able to cut with a straight line through a volume of material, so each cut creates a ruled surface (or more specifically, one of a limited subset of ruled surfaces--no Moebius strips or Pluecker's Conoids allowed).
What volumes can be formed from a combination of such cuttable surfaces? It seems fairly intuitive that because of the cutter's limitations, the shapes must be continuous, non-self-intersecting, and holeless. There is also some sort of convexity requirement, thought I'm not sure exactly how to phrase it--I don't know a mathematically strict definition of concavity and curvature that would categorize all the shapes you can cut--hypars, hyperboloids, etc--together. I'm pretty sure the answer is in here somewhere, if only I could read it.
Also, doubly-curved surfaces--a sphere, for example--meet all of the above criteria, but you still don't want to cut them with one of these devices because they can only be approximated, and then only inefficiently (too many cuts).
At any rate, it seems that this whittles the set of available shapes down a little more than I'd like, in terms of architectural potential. That means chopping up multiple bits of foam and sticking them together into more complex shapes.
So the next question becomes: what shapes can be subdivided into multiple cuttable volumes whose interfaces are cuttable surfaces?
Paul Haeberli's Lamina is basically solving a related problem in 2d surfaces, instead of solid volumes.
There's got to be someone working to develop efficient algorithms for breaking down arbitrary 3d volumes into the most efficient set of cuttable volumes. It's way over my head at this point, but a super-interesting problem...