**Reference**:- Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams, “Rolling Polyhedra on Tessellations”, in
*Proceedings of the 11th International Conference on Fun with Algorithms (FUN 2022)*, Island of Favignana, Sicily, Italy, May 30–June 3, 2022, 5:1–5:16. **Abstract**:-
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. **Availability**:- The paper is available in PDF (2971k).
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