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DOI: 10.4230/LIPIcs.FUN.2022.6

Rolling Polyhedra on Tessellations

Authors: Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

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.

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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)

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@InProceedings{baes_et_al:LIPIcs.FUN.2022.6,
  author =	{Baes, Akira and Demaine, Erik D. and Demaine, Martin L. and Hartung, Elizabeth and Langerman, Stefan and O'Rourke, Joseph and Uehara, Ryuhei and Uno, Yushi and Williams, Aaron},
  title =	{{Rolling Polyhedra on Tessellations}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.6},
  URN =		{urn:nbn:de:0030-drops-159761},
  doi =		{10.4230/LIPIcs.FUN.2022.6},
  annote =	{Keywords: polyhedra, tilings}
}

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Rolling Polyhedra on Tessellations

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