Nonsense! Your post was great fun. The code was fun, too. I particularly enjoyed your comment about to how Pi was referenced in the bible, along with the ensuing rabbit trail of research that took me down.[1]
You could lengthen the bridge (theoretically, the pieces might not fit that snug) with any combination of the pieces all the way down to the case of a reverse bridge connected at the bottom.
Won't their definition conflict with the rectilinear definition? Specifically, there are distinct rectilinear configurations which are homotopic. With two bricks connected by at corner, there are two rectilinear configurations but both are homotopic. So their method used to count "homotopy equivalence classes which contain at least one rectilinear configuration" would yield a lower number than the previous result. This seems to contradict when the paper says that their definition extends the previous one.
Where are you getting two from? By the definitions on this page, it seems to me like there are four corner-connected rectilinear configurations of two bricks, and also four homotopy equivalence classes, so there's no inconsistency.
I've always wanted to make a sorting box that, given any quantity of random Lego Mindstorms parts, sorts the parts into shape/size/purpose bins. A static box, no moving parts but the raw feed.
I suppose it will have to be a big box, mathematically.
http://blog.jgc.org/2010/01/more-fun-with-toys-ikea-lillabo-...