Research Article
Year 2022,
Volume: 15 Issue: 2, 175 - 177, 31.10.2022
Abstract
In 1930, Keller conjectured that every tiling of RnRn by unit cubes contains a pair of cubes sharing a complete (n−1)(n−1)-dimensional face. Only 50 years later, Lagarias and Shor found a counterexample for all n≥10n≥10. In this note we show that neither a modification of Keller's conjecture to tiles of more complex shape is true.
References
- Brakensiek, J., Heule, M., Mackey, J., Narvaez, D.: The Resolution of Keller's Conjecture. In: International Joint Conference on Automated Reasoning (IJCAR 2020), 48–65 (2020). https://doi.org/10.1007/978-3-030-51074-9_4
- Debroni, J., Eblen, J. B., Langston, M. A., Myrvold, W., Shor, P., Weerapurage, D.: A complete resolution of the Keller maximum clique problem. In: Proceedings of the Twenty-Second Annual ACMSIAM Symposium on Discrete Algorithms (SODA 11), 129–135 (2011). https://doi.org/10.1137/1.9781611973082.11
- Hajos, G.: Uber einfache und mehrfache Bedeckung des n-dimensional Raumes mit einem Wurfelgitter. Math. Zeitschr. 47, 427–467 (1942).
- Keller, O. H.: Uber die luckenlose Einfullung des Raumes mit Wurfeln. J. Reine Angew. Math 177, 231–248 (1930).
- Lagarias, J. F., Shor, P. W.: Keller’s cube-tiling conjecture is false in high dimensions. Bull. Amer. Math. Soc. 27, 279–283 (1992). https://doi.org/10.1090/S0273-0979-1992-00318-X
- Lysakowska, M., Przeslawski, K.: Keller’s conjecture on the existence of columns in cube tiling of Rn. Adv. Geom. 12 (2), 329–352 (2012). https://doi.org/10.1515/advgeom.2011.055
- Mackey, J.: Cube tiling of dimension eight with no facesharing. Discrete & Computational Geometry 28, 275–279 (2002). https://doi.org/10.1007/s00454-002-2801-9
- Minkowski, H.: Dichtestegittenformige Lagerung kongruenter Korper. Nachrichten Ges. Wiss. Gottingen, 311–355 (1904).
- Perron, O.: Modulartige luckenlose Ausfullung des Rn mit kongruente Wurfeln I. Math. Ann., 415–447 (1940).
Year 2022,
Volume: 15 Issue: 2, 175 - 177, 31.10.2022
Abstract
References
- Brakensiek, J., Heule, M., Mackey, J., Narvaez, D.: The Resolution of Keller's Conjecture. In: International Joint Conference on Automated Reasoning (IJCAR 2020), 48–65 (2020). https://doi.org/10.1007/978-3-030-51074-9_4
- Debroni, J., Eblen, J. B., Langston, M. A., Myrvold, W., Shor, P., Weerapurage, D.: A complete resolution of the Keller maximum clique problem. In: Proceedings of the Twenty-Second Annual ACMSIAM Symposium on Discrete Algorithms (SODA 11), 129–135 (2011). https://doi.org/10.1137/1.9781611973082.11
- Hajos, G.: Uber einfache und mehrfache Bedeckung des n-dimensional Raumes mit einem Wurfelgitter. Math. Zeitschr. 47, 427–467 (1942).
- Keller, O. H.: Uber die luckenlose Einfullung des Raumes mit Wurfeln. J. Reine Angew. Math 177, 231–248 (1930).
- Lagarias, J. F., Shor, P. W.: Keller’s cube-tiling conjecture is false in high dimensions. Bull. Amer. Math. Soc. 27, 279–283 (1992). https://doi.org/10.1090/S0273-0979-1992-00318-X
- Lysakowska, M., Przeslawski, K.: Keller’s conjecture on the existence of columns in cube tiling of Rn. Adv. Geom. 12 (2), 329–352 (2012). https://doi.org/10.1515/advgeom.2011.055
- Mackey, J.: Cube tiling of dimension eight with no facesharing. Discrete & Computational Geometry 28, 275–279 (2002). https://doi.org/10.1007/s00454-002-2801-9
- Minkowski, H.: Dichtestegittenformige Lagerung kongruenter Korper. Nachrichten Ges. Wiss. Gottingen, 311–355 (1904).
- Perron, O.: Modulartige luckenlose Ausfullung des Rn mit kongruente Wurfeln I. Math. Ann., 415–447 (1940).
There are 9 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | July 23, 2022 |
| Publication Date | October 31, 2022 |
| Acceptance Date | May 20, 2022 |
| Published in Issue | Year 2022 Volume: 15 Issue: 2 |
