Trajectoid
A trajectoid is a geometric shape designed to trace a predetermined general periodic trajectory on the plane.[1] Unlike more familiar rolling bodies such as spheres or cylinders, which follow linear or sinusoidal paths, trajectoids can trace even very complex, custom trajectories. The concept extends earlier studies on rolling objects like sphericons and oloids, introducing much greater diversity in possible rolling paths.
Physicists and mathematicians from the Institute for Basic Science, Ulsan National Institute of Science and Technology and the University of Geneva, developed an algorithm that defines 3D shapes that will roll under gravity along a general periodic trajectory on an inclined plane (or equivalently can be rolled by hand on a horizontal plane).[2] This algorithm was realized through 3D printing: Trajectoids were produced by printing two halves and a spherical hollow center which would contain a metal sphere for mass.
The authors conjectured that trajectoids can be generated for any planar periodic trajectory as long as the trajectoid returns to its original orientation after two or more periods.[3] This conjecture was later proved as part of a more general theorem that applies to any system governed by the rotation group SO(3) or the unitary group SU(2),[4] such as spins, qubits and gyroscopes. The theorem states any series of rotations operating on such a physical system (except for a zero-measure set) can be made to return the system to its original state by uniformly scaling and repeating the series twice. The existence of trajectoids follows from Minkowski's theorem and the Rodrigues formula for the product of rotation matrices.
References
- ↑ Sobolev, Yaroslav I.; Dong, Ruoyu; Tlusty, Tsvi; Eckmann, Jean-Pierre; Granick, Steve; Grzybowski, Bartosz A. (2023). "Solid-body trajectoids shaped to roll along desired pathways". Nature 620 (7973): 310–315. doi:10.1038/s41586-023-06306-y. PMID 37558849. Bibcode: 2023Natur.620..310S. https://www.nature.com/articles/s41586-023-06306-y.
- ↑ Eckmann, Jean-Pierre; Sobolev, Yaroslav I.; Tlusty, Tsvi (2024). "Tumbling Downhill Along a Given Curve". Notices of the American Mathematical Society 71 (6): 740–747. doi:10.1090/noti2947.
- ↑ Matsumoto, Elisabetta; Segerman, Henry (2023). "Shaped to roll along a programmed periodic path" (in en). Nature 620 (7973): 282–283. doi:10.1038/d41586-023-02335-9. https://www.nature.com/articles/d41586-023-02335-9.
- ↑ Eckmann, Jean-Pierre; Tlusty, Tsvi (2025-10-01). "Walks in Rotation Spaces Return Home when Doubled and Scaled" (in en). Physical Review Letters 135 (14). doi:10.1103/xk8y-hycn. ISSN 0031-9007. https://link.aps.org/doi/10.1103/xk8y-hycn.
- Wonky shapes can be made to roll along a specific path of your choice Video by New Scientist
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