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Licensed Unlicensed Requires Authentication Published by De Gruyter October 13, 2016

Optimal cover of a disk with three smaller congruent disks

Balázs Szalkai
From the journal Advances in Geometry

Abstract

In 2008 R. Connelly asked how one should place n small disks of radius r to cover the largest possible area of a disk of radius R > r. More specifically, is there always an optimal configuration with n-fold rotational symmetry for small values of n? The answer is known to be positive for n = 2, negative for n = 5, and it has been conjectured to be positive for n = 3 and 4. In this paper, we present a systematic way to list all possible combinatorial structures of optimal configurations, and we prove that for n = 3 there is always an optimal configuration with rotational symmetry of order three.

MSC 2010: 05B40; 52C15

Communicated by: G. Korchmáros


References

[1] K. Bezdek, Optimal covering of circles. PhD thesis, Eötvös University, 1979.Search in Google Scholar

[2] K. Bezdek, Über einige Kreisüberdeckungen. Beiträge Algebra Geom. 14 (1983), 7–13. MR695880 (85a:52012) Zbl 0507.52009Search in Google Scholar

[3] B. Csikós, On the volume of the union of balls. Discrete Comput. Geom. 20 (1998), 449–461. MR1651904 (99g:52008) Zbl 0922.5101010.1007/PL00009395Search in Google Scholar

[4] B. Csikós, On the volume of flowers in space forms. Geom. Dedicata86 (2001), 59–79. MR1856418 (2002f:51029) Zbl 0990.5101010.1023/A:1011983123985Search in Google Scholar

[5] B. Csikós, A Schläfli-type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture. Monatsh. Math. 147 (2006), 273–292. MR2215837 (2007a:53078) Zbl 1093.5304110.1007/s00605-005-0363-7Search in Google Scholar

[6] F. Fodor, The densest packing of 19 congruent circles in a circle. Geom. Dedicata74 (1999), 139–145. MR16740 49 (2000f:52022) Zbl 0927.5202410.1023/A:1005091317243Search in Google Scholar

[7] F. Fodor, The densest packing of 12 congruent circles in a circle. Beiträge Algebra Geom. 41 (2000), 401–409. MR1801430 (2001k:52037) Zbl 0974.52015Search in Google Scholar

[8] F. Fodor, The densest packing of 13 congruent circles in a circle. Beiträge Algebra Geom. 44 (2003), 431–440. MR2017043 (2004i:52016) Zbl 1039.52015Search in Google Scholar

[9] F. Fodor, Packing 14 congruent circles in a circle. Stud. Univ. Žilina Math. Ser. 16 (2003), 25–34. MR2065745 (2005b:52043) Zbl 1057.52010Search in Google Scholar

[10] Z. Gáspár, T. Tarnai, K. Hincz, Bifurcations of an elastic model with nonsmooth material law. In: IV European Conference on Computational Mechanics, Paris, France, 2010.Search in Google Scholar

[11] R. L. Graham, Sets of points with given minimum separation. Solution to Problem E1921. Amer. Math. Monthly75 (1968), 192–193.10.2307/2315913Search in Google Scholar

[12] H. Melissen, Densest packings of eleven congruent circles in a circle. Geom. Dedicata50 (1994), 15–25. MR1280791 (95e:52032) Zbl 0810.5201310.1007/BF01263647Search in Google Scholar

[13] U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten. Math. Nachr. 40 (1969), 111–124. MR0253164 (40 #6379) Zbl 0182.2510210.1002/mana.19690400110Search in Google Scholar

[14] G. Fejes Tóth, Thinnest covering of a circle by eight, nine, or ten congruent circles. In: Combinatorial and computational geometry, volume 52 of Math. Sci. Res. Inst. Publ., 361–376, Cambridge Univ. Press 2005. MR2178327 (2007b:52028) Zbl 1097.52006Search in Google Scholar

Received: 2013-10-14
Revised: 2015-4-1
Published Online: 2016-10-13
Published in Print: 2016-10-1

© 2016 by Walter de Gruyter Berlin/Boston

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