Critical configurations of solid bodies and the Morse theory of MIN functions
- Authors: Ogievetskii O.V.1,2,3, Shlosman S.B.1,4,5
-
Affiliations:
- Aix-Marseille Université
- P. N. Lebedev Physical Institute of the Russian Academy of Sciences
- Kazan (Volga Region) Federal University
- Institute for Information Transmission Problems, Russian Academy of Sciences
- Skolkovo Institute of Science and Technology
- Issue: Vol 74, No 4 (2019)
- Pages: 59-86
- Section: Articles
- URL: https://journals.rcsi.science/0042-1316/article/view/133563
- DOI: https://doi.org/10.4213/rm9899
- ID: 133563
Cite item
Open Access
Access granted
Subscription Access
Abstract
This paper studies the manifold of clusters of non-intersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. The notion of critical cluster is introduced, and several critical clusters of balls and of cylinders are studied. In the case of cylinders, some of the critical clusters here are new. The paper also establishes criticality properties of clusters introduced earlier by Kuperberg [7].
About the authors
Oleg Viktorovich Ogievetskii
Aix-Marseille Université; P. N. Lebedev Physical Institute of the Russian Academy of Sciences; Kazan (Volga Region) Federal University
Email: oleg@cpt.univ-mrs.fr
Candidate of physico-mathematical sciences
Semen Bensionovich Shlosman
Aix-Marseille Université; Institute for Information Transmission Problems, Russian Academy of Sciences; Skolkovo Institute of Science and Technology
Email: shlosman@cpt.univ-mrs.fr
Doctor of physico-mathematical sciences
References
- R. Buckminster Fuller, E. J. Applewhite, Synergetics. Explorations in the geometry of thinking, MacMillan Co., New York, 1975, xxxii+876 pp.
- J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss., 290, 3rd ed., Springer-Verlag, New York, 1999, lxxiv+703 pp.
- H. S. M. Coxeter, Regular polytopes, 3rd ed., Dover Publications, Inc., New York, 1973, xiv+321 pp.
- L. Fejes Toth, Lagerungen in der Ebene auf der Kugel und im Raum, Grundlehren Math. Wiss., 65, 2. verb. und erweit. Aufl., Springer-Verlag, Berlin–New York, 1972, xi+238 pp.
- M. Firsching, Optimization methods in discrete geometry, Dissertation, Freie Universität, Freie Univ., Berlin, 2016, 85 pp., par
- A. Heppes, L. Szabo, “On the number of cylinders touching a ball”, Geom. Dedicata, 40:1 (1991), 111–116
- W. Kuperberg, “How many unit cylinders can touch a unit ball? (Problem 3.3)”, DIMACS Workshop on polytopes and convex sets, Rutgers Univ., 1990
- R. Kusner, W. Kusner, J. C. Lagarias, S. Shlosman, “Configuration spaces of equal spheres touching a given sphere: the twelve spheres problem”, New trends in intuitive geometry, Bolyai Soc. Math. Stud., 27, Janos Bolyai Math. Soc., Budapest, 2018, 219–277
- J. C. Lagarias (ed.), The Kepler conjecture. The Hales–Ferguson proof, Springer, New York, 2011, xiv+456 pp.
- O. Ogievetsky, S. Shlosman, “The six cylinders problem: $mathbb D_{3}$-symmetry approach”, Discrete Comput. Geom., publ. online 2019, 1–20
- O. Ogievetsky, S. Shlosman, Extremal cylinder configurations I: Configuration $C_{mathfrak{m}}$, 2018, 38 pp.
- O. Ogievetsky, S. Shlosman, Extremal cylinder configurations II: Configuration $O_{6}$, 2019, 25 pp.
- O. Ogievetsky, S. Shlosman, Platonic compounds of cylinders, 2019, 35 pp.
Supplementary files
