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Extremal cylinder configurations. II: Configuration \(O_6\). (English) Zbl 07566897

Summary: We study the octahedral configuration \(O_6\) [Kuperberg] of six equal cylinders touching the unit sphere. We show that the configuration \(O_6\) is a local sharp maximum of the distance function. Thus, it is not unlockable and, moreover, rigid. For Part I, see [O. Ogievetsky and S. Shlosman, Discrete Comput. Geom. 66, No. 1, 140–164 (2021; Zbl 1468.52012)].

MSC:

51-XX Geometry
30-XX Functions of a complex variable

Citations:

Zbl 1468.52012

Software:

Mathematica
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Full Text: DOI arXiv

References:

[1] Conway, J. H.; Sloane, N. J. A., Sphere Packings, Lattices and Groups, 290 (2013), New York: Springer Science & Business Media, New York
[2] Kuperberg, W. (1990)
[3] Kuperberg, W., MathOverflow page
[4] Kusner, R.; Kusner, W.; Lagarias, J. C.; Shlosman, S.; Ambrus, G.; Bárány, I.; Böröczky, K.; Fejes Tóth, G.; Pach, J., New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, 27, Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem (2018), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 1411.52002
[5] [Ogievetsky and Shlosman 18] Ogievetsky, O. and Shlosman, S.. “Extremal Cylinder Configurations I: Configuration \(####\)” arXiv:1812.09543, 2018. · Zbl 1468.52012
[6] [Ogievetsky and Shlosman 19a] Ogievetsky, O. and Shlosman, S.. “The Six Cylinders Problem: \(####\)-Symmetry Approach.” Discrete & Computational Geometry (2019a), 1-20. · Zbl 1458.05041
[7] [Ogievetsky and Shlosman 19b] Ogievetsky, O. and Shlosman, S.. “Platonic Compounds of Cylinders.” arXiv:1904.02043, 2019b. · Zbl 1468.52013
[8] [Viro and Viro 06] Viro, J. and Viro, O.. “Configurations of Skew Lines.” arXiv preprint math/0611374, 2006. · Zbl 1131.14062
[9] Wolfram Research, Inc, Mathematica, Version 11.3 (2018), Champaign, IL: Wolfram Research, Inc, Champaign, IL
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