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].
Keywords:
configurations of balls, configurations of cylinders, rigid clusters, flexible clusters, critical clusters, connected components, Galois symmetries, Platonic configurations, maxima of non-analytic functions.
Part of the work of S.B. Shlosman was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences and was supported by the Russian Science Foundation (project no. 14-50-00150).
Another part was carried out in the framework of the Labex Archimède programme (grant ANR-11-LABX-0033) and the A*MIDEX project (grant ANR-11-IDEX-0001-02), funded by the ‘Investissements d’Avenir' French Government programme managed by the French National Research Agency (ANR). The work of the O.V. Ogievetsky was supported by the Competitive Growth Programme of Kazan Federal University (project “5-100”) and by the Russian Foundation for Basic Research (grant no. 17-01-00585).
Citation:
O. V. Ogievetsky, S. B. Shlosman, “Critical configurations of solid bodies and the Morse theory of MIN functions”, Russian Math. Surveys, 74:4 (2019), 631–657
Citation in format AMSBIB
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