Document Open Access Logo

Pach’s Animal Problem Within the Bounding Box

Author Martin Tancer

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.78.pdf
  • Filesize: 1.68 MB
  • 18 pages

Document Identifiers

Author Details

Martin Tancer
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

Supported by the GAČR grant no. 22-19073S.

Cite AsGet BibTex

Martin Tancer. Pach’s Animal Problem Within the Bounding Box. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 78:1-78:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.78

Abstract

A collection of unit cubes with integer coordinates in ℝ³ is an animal if its union is homeomorphic to the 3-ball. Pach’s animal problem asks whether any animal can be transformed to a single cube by adding or removing cubes one by one in such a way that any intermediate step is an animal as well. Here we provide an example of an animal that cannot be transformed to a single cube this way within its bounding box.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Animal problem
  • bounding box
  • non-shellable balls

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Zachary Abel and Scott D Kominers. Universal reconfiguration of (hyper-) cubic robots. arXiv preprint, 2008. URL: https://arxiv.org/abs/0802.3414.
  2. Karim A. Adiprasito and Bruno Benedetti. Subdivisions, shellability, and collapsibility of products. Combinatorica, 37(1):1-30, 2017. URL: https://doi.org/10.1007/s00493-016-3149-8.
  3. Hugo A. Akitaya, Esther M. Arkin, Mirela Damian, Erik D. Demaine, Vida Dujmović, Robin Flatland, Matias Korman, Belen Palop, Irene Parada, André van Renssen, and Vera Sacristán. Universal reconfiguration of facet-connected modular robots by pivots: the O(1) musketeers. Algorithmica, 83(5):1316-1351, 2021. URL: https://doi.org/10.1007/s00453-020-00784-6.
  4. R. H. Bing. Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. In Lectures on Modern Mathematics, Vol. II, pages 93-128. Wiley, New York, 1964. Google Scholar
  5. Adrian Dumitrescu and Evan Hilscher. Animal testing. In Algorithms and Computation - 22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings, volume 7074 of Lecture Notes in Comput. Sci., pages 220-229. Springer, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-25591-5_24.
  6. Adrian Dumitrescu and János Pach. Pushing squares around. In Proceedings of the twentieth annual Symposium on Computational geometry, pages 116-123, 2004. URL: https://doi.org/10.1007/s00373-005-0640-1.
  7. Daniel Feshbach and Cynthia Sung. Reconfiguring non-convex holes in pivoting modular cube robots. IEEE Robotics and Automation Letters, 6(4):6701-6708, 2021. URL: https://doi.org/10.1109/LRA.2021.3095030.
  8. Robert Furch. Zur grundlegung der kombinatorischen topologie. Abh. Math. Sem. Univ. Hamburg, 3(1):69-88, 1924. URL: https://doi.org/10.1007/BF02954617.
  9. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  10. Reinhard Klette and Azriel Rosenfeld. Digital geometry. Morgan Kaufmann Publishers, San Francisco, CA; Elsevier Science B.V., Amsterdam, 2004. Geometric methods for digital picture analysis. Google Scholar
  11. Irina Kostitsyna, Tim Ophelders, Irene Parada, Tom Peters, Willem Sonke, and Bettina Speckmann. Optimal in-place compaction of sliding cubes. arXiv preprint, 2023. URL: https://arxiv.org/abs/2312.15096.
  12. Akira Nakamura. A solution to the Animal Problem. Department of Computer Science, University of Auckland, 2010. URL: https://cerv.aut.ac.nz/wp-content/uploads/2015/08/MItech-TR-53.pdf.
  13. Joseph O'Rourke. Pach’s "animals": What if the genus is positive? MathOverflow question. URL: https://mathoverflow.net/questions/50966/pachs-animals-what-if-the-genus-is-positive.
  14. Joseph O'Rourke. The computational geometry column# 4. ACM SIGGRAPH Computer Graphics, 22(2):111-112, 1988. Google Scholar
  15. C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
  16. Thomas Shermer. A smaller irreducible animal; and a very small irreducible animal. Snapshots of Computational and Discrete Geometry, pages 139-143, 1988. Google Scholar
  17. Martin Tancer. Pach’s animal problem within the bounding box. arXiv preprint, 2024. URL: https://arxiv.org/abs/2402.18212.
  18. J. H. C. Whitehead. Simple homotopy types. Amer. J. Math., 72:1-57, 1950. URL: https://doi.org/10.2307/2372133.
  19. G. M. Ziegler. Shelling polyhedral 3-balls and 4-polytopes. Discrete Comput. Geom., 19(2):159-174, 1998. URL: https://doi.org/10.1007/PL00009339.
  20. Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. URL: https://doi.org/10.1007/978-1-4613-8431-1.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact