UC Berkeley postdoc Aaron M Hoover combined math and imagination to solve the problem of building a one-sided 'Möbius gear.' He rendered it and then output molds for it on a 3D printer, cast them, and assembled his freaky, mind-melting beast.
While searching for a suitable project for CS 285 (Procedural Solid Modeling) I was introduced to the Möbius gear by Professor Sequin. I was immediately intrigued by the curious combination of the Möbius mathematical surface popularized by M.C. Escher and functional mechanical gear elements. After some time staring at and puzzling over this image, I convinced myself that this mechanism is indeed possible and that with right tools, a functional prototype could be built. (The entire mechanism essentially boils down to an oddly configured set of planetary gears. One can think of the black portion in the image as the ring with a fixed zero input velocity. A single blue gear is a planet, and the white strip is the sun. Output can be taken either from the sun or the planets (with no regard for practicality!). In practice, however, it's easiest to actuate the Möbius strip (the white portion).
The Möbius Gear
(via Neatorama)
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