Abstract
We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place n static “balls” with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is \(\sqrt{n}\) (up to constants). The lower bound is based on the Erdős-Szekeres Theorem. The upper bound may be considered a generalization of the Erdős-Szekeres Theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, Vladimir I.: Vladimir I. Arnold—collected works. Vol. II.Hydrodynamics, bifurcation theory, and algebraic geometry1965–1972. Springer-Verlag, Berlin, 2014. Edited by Alexander B.Givental, Boris A. Khesin, Alexander N. Varchenko, Victor A.Vassiliev and Oleg Ya. Viro
Athreya, Jayadev., Burdzy, Krzysztof., Duarte, Mauricio.: On pinned billiard balls and foldings. Math (2018). arXiv:1807.08320
Blackwell, Paul.: An alternative proof of a theorem of Erdös and Szekeres. Amer. Math. Monthly 78, 273 (1971)
Burago, D., Ferleger, S., Kononenko, A.: A geometric approach to semi-dispersing billiards. Ergodic Theory Dynam. Systems 18(2), 303–319 (1998)
Burago, D., Ferleger, S., Kononenko, A.: Unfoldings and global bounds on the number of collisions for generalized semi-dispersing billiards. Asian J. Math. 2(1), 141–152 (1998)
Burago, D., Ferleger, S., Kononenko, A.: Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. (2), 147(3), 695–708 (1998)
Burago, D., Ferleger, S., Kononenko, A.: A geometric approach to semi-dispersing billiards. In Hard ball systems and the Lorentz gas, volume 101 of Encyclopaedia Math. Sci., pages 9–27. Springer, Berlin, (2000)
Burago, D. Ferleger, S. Kononenko, A.: Collisions in semi-dispersing billiard on Riemannian manifold. In Proceedings of the International Conference on Topology and its Applications (Yokohama, 1999), volume 122, pages 87–103, (2002)
Burago, Dmitri., Ivanov, Sergei.: Examples of exponentially many collisions in a hard ball system. arXiv e-prints , September Math arXiv:1809.02800 (2018)
Burdzy, Krzysztof, Duarte, Mauricio: A lower bound for the number of elastic collisions. Communications in Mathematical Physics (2019). https://doi.org/10.1007/s00220-019-03399-3
Chen, Xinfu, Illner, Reinhard: Finite-range repulsive systems of finitely many particles. Arch. Ration. Mech. Anal. 173(1), 1–24 (2004)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 463–470 (1935)
Illner, Reinhard: On the number of collisions in a hard sphere particle system in all space. Transport Theory Statist. Phys. 18(1), 71–86 (1989)
Illner, Reinhard.: Finiteness of the number of collisions in a hard sphere particle system in all space. II. Arbitrary diameters and masses. Transport Theory Statist. Phys., 19(6), 573–579 (1990)
Seidenberg, A.: A simple proof of a theorem of Erdös and Szekeres. J. London Math. Soc. 34, 352 (1959)
Vaserstein, L.N.: On systems of particles with finite-range and/or repulsive interactions. Comm. Math. Phys. 69(1), 31–56 (1979)
Acknowledgements
We are grateful to the anonymous referees for suggestions for improvement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
JSA’s research was supported in part by NSF CAREER grant DMS 1559860. KB’s research was supported in part by Simons Foundation Grant 506732.
Rights and permissions
About this article
Cite this article
Athreya, J., Burdzy, K. Protecting Billiard Balls From Collisions. Arnold Math J. 6, 57–62 (2020). https://doi.org/10.1007/s40598-019-00131-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-019-00131-w