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# Rolling Polyhedra on Tessellations

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LIPIcs.FUN.2022.6.pdf
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• 16 pages

## Acknowledgements

Part of this work appeared in the first author’s Master’s Thesis. Part of this work was done at the 1st and 2nd Virtual Workshops on Computational Geometry (2020 and 2021). The authors would like to thank all participants of those workshops. Renders of prisms and antiprisms are by Robert Webb’s http://www.software3d.com/Stella.php. Other polyhedron renders are from Wikimedia Commons under Creative Commons Attribution license.

## Cite As

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 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FUN.2022.6

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Theory of computation → Design and analysis of algorithms
• Mathematics of computing → Discrete mathematics
• polyhedra
• tilings

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## References

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