In preparation for doing some 3D printing & designing, I've been learning about structural engineering & trusses, and tripped over a very beautiful part of mathematics. As an electrical engineering student at MIT, I learned about planar & dual graphs for circuit diagrams, but the duality is even stronger & prettier for the 'trusses' found in civil & mechanical engineering. For example, if you have a planar system of rods (of fixed lengths) and (pin) joints which make up a 'truss', you can analyze this diagram to compute internal forces when external forces are applied to this structure. The cool thing about these truss diagrams is that the _dual diagram_ (in the graph theory sense) represents precisely the directions and magnitudes (lengths of the arrows) of the forces in the primal diagram (and vice versa). This duality was discovered by Maxwell (of Maxwell's equations). Thus, whereas in electrical engineering the dual diagram was purely conceptual; in trusses, the dual diagram has the actual x,y components of the forces. Structural engineers for about 50 years after Maxwell used these dual diagrams -- called 'graphic statics' -- to design many bridges, buildings, etc. (I would even speculate that Archimedes may have intuitively understood this duality, but we don't have enough extant material from Archimedes to know for sure.) http://en.wikipedia.org/wiki/Cremona_diagram Michell, following up on Maxwell, came out with 'Michell trusses' which optimized the amount of material required to carry a particular load. Some of Michell's structures look like solutions of Maxwell's (electromagnetic) Equations, and indeed they are! One of Michell's solution form logarithmic spirals in the complex plane; one can generate some of these discrete spirals using Gaussian integer multiplication -- including Gaussian prime numbers. Even cooler, the discrete form of some of Michell's trusses are _self-reciprocal_ (self-dual), in the sense that the force diagram and the truss diagram are identical (although the 'force' edge doesn't usually correspond with itself, but with another edge in another part of the same diagram). But wait, there's more! AI vision folks in the 1960's & 1970's were trying to 'see' & interpret simple objects like polyhedra using video cameras, and rediscovered another portion of Maxwell's theory: that _projections_ of polyhedra onto planes had certain combinatorial constraints; and Maxwell found that these constraints were identical to those involving stresses in planar 2D truss networks. Here's the basic intuition, which has be formalized into a proof. Consider a (3D) tetrahedron and its projection onto the plane (assume that no face is perpendicular to the plane, and no vertices or edges coincide). You get a 2D diagram with 4 vertices, 6 edges, and 4 'windows' -- including the region outside the diagram. Consider that the tetrahedron as a set of 6 bars/edges connecting the 4 vertices in 3D. If you _stand on_ the tetrahedron, the bottom 3 edges/bars will be in _tension_, while the 3 edges/bars holding up the 4 vertex will be in _compression_. Alternatively, if you look at the 2D projection of the tetrahedron, and attempt to _pick it up using the center vertex_, this process will induce _tension_ in the middle 3 edges/bars, and _compression_ in the outer 3 edges/bars. Thus, there is a perfect correspondence between 2D diagrams which can support 'self-stress' (through tension & compression of the bars/edges) and projections of 3D polyhedra. Interestingly, Maxwell knew about this 3D polyhedra/2D truss correspondence, too! --- I believe that there may be even more reciprocal/dual beauty hidden in the control points of Bezier splines and 'forces' on the splines, but I haven't gotten that far yet. For example, a typical suspension bridge provides _piers_, cables and suspenders to keep a roadway horizontal in the presence of loads, but if you want a _curved_ roadway, you need multiple piers, cables & suspenders to hold that roadway in the shape of the curve. The 'piers' in this case act like 'control points' for the (curved) roadway. http://en.wikipedia.org/wiki/De_Casteljau's_algorithm --- Besides Maxwell & Michell, there are many, many other names: Huffmann, Laman, Crapo, Whiteley, Sugihara, etc.