Still: http://gosper.org/pumpstill.gif I noted the 2D porism in high school after reading an article on Wankel engines. In 1965 I turned it into a "display hack" for the PDP-6 using uncorrected Minsky "circles" for both the shapes (curved dotted lines) and the motion. The eccentricities were not noticeable. As soon as Minsky saw it, he exclaimed "Helical twist!". Prior to that, I had tried to make it into a pump or an engine with various impractical kludges which I later learned were patented in the 1940s. I then devised the continuous deformation via "four point ellipses" to circles as a means of porting the intake and exhaust, and we always mentioned the twist and taper when we showed AILab visitors the crude 2D demo. I later learned that the twist idea without the taper to circularity was patented by someone in Florida a few months later. I've always suspected, by someone who stole the idea but didn't understand the taper description, which was difficult in those days to draw and very difficult to convey without a drawing, as I later learned by wasting several hundred dollars on inaccurate patent searches. When I first showed the 2D porism to Rich, he immediately remarked that three "Wankels" (Reuleaux triangles) do it too. Two other improvements which seemed unpatented at the time were to vary the pitch along the length to get an arbitrary compression profile (e.g., for Diesels), and to tessellate six or more rotors to make equally many pumping cells in opposite phases, which turns out deliver a constant, nonpulsing dVolume/dt (neat proof!) Alternatively, a single set of four blunt rotors could replace the "Ferris wheel" of rollers munching the compressible hose in a heart-lung machine. And I've always suspected that a sheet of small, low-friction rotors would form a "positive displacement wall" if driven just by the rotors along the edges, providing high volume and high pressure, but low velocity. With cones based on Lissajous arcs, it is even possible to mesh nonparallel rotors. I was never able to visualize the constant shape pumped by four fixed pitch rotors. Its ends are like twisted square pyramids with concave sides, but "bookending" them ought to approximate an octahedron, which is impossible, since there are only four smooth sides. Julian Ziegler Hunts to the rescue: http://gosper.org/pumpee.png . This is what provoked my msg about rwg> Good grief, tutor Julian has plotted two infinitely differentiable
surfaces intersecting in a simple arc that is not differentiable.
Julian also determined the maximum removable rotor surface (after initially doubting that any reduction was possible): http://gosper.org/minpump.png His animation of four reduced rotors pumping a string of pumpees is weird. --rwg