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Rolling Polyhedra on Tessellations

par Baes, Akira;Demaine, Erik D. ;Demaine, Martin L.;Hartung, Elizabeth;Langerman, Stefan ;O'Rourke, Joseph ;Uehara, Ryuhei;Uno, Yushi;Williams, Aaron
Référence 11th International Conference on Fun with Algorithms, FUN 2022, May 30 to June 3, 2022, Island of Favignana, Sicily, Italy 11th International Conference on Fun with Algorithms, FUN 2022, May 30 to June 3, 2022, Island of Favignana, Sicily, Italy, Schloss Dagstuhl - Leibniz-Zentrum f"ur Informatik, Vol. 226, page (6:1-6:16)
Publication Publié, 2022

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Résumé :

We study the space reachable by rolling a 3D convex polyhedron on a 2D periodic tessellation in the xy-plane, where at every step a face of the polyhedron must coincide exactly with a tile of the tessellation it rests upon, and the polyhedron rotates around one of the incident edges of that face until the neighboring face hits the xy plane. If the whole plane can be reached by a sequence of such rolls, we call the polyhedron a plane roller for the given tessellation. We further classify polyhedra that reach a constant fraction of the plane, an infinite area but vanishing fraction of the plane, or a bounded area as hollow-plane rollers, band rollers, and bounded rollers respectively. We present a polynomial-time algorithm to determine the set of tiles in a given periodic tessellation reachable by a given polyhedron from a given starting position, which in particular determines the roller type of the polyhedron and tessellation. Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on every regular-tiled (≤ 4)-uniform tessellation.