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Packing 13 circles in an equilateral triangle

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Abstract

The maximum separation problem is to find the maximum of the minimum pairwise distance of n points in a planar body \({\mathcal {B}}\) on the Euclidean plane. In this paper this problem will be considered if \({\mathcal {B}}\) is the equilateral triangle of side length 1 and the number of points is 13. We will present the exact separation distance of 13 points in the equilateral triangle of side length 1 and we will prove a conjecture of Melissen from 1993 and a conjecture of Graham and Lubachevsky from 1995.

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References

  1. Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)

    MATH  Google Scholar 

  2. Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, Berlin (1991)

    Book  Google Scholar 

  3. Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin (1972)

    Book  Google Scholar 

  4. Goodman, J.E., O’Rourke, J., Tóth, CsD: Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton (2018)

    MATH  Google Scholar 

  5. Graham, R.L., Lubachevsky, B.D.: Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond. Electron. J. Comb. 2, A1 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Groemer, H.: Zusammenhängende Lagerungen konvexer Körper. Math. Z. 94, 66–78 (1966)

    Article  MathSciNet  Google Scholar 

  7. Melissen, J.B.M.: Densest packings of congruent circles in an equilateral triangle. Am. Math. Mon. 100, 816–825 (1993)

    Article  MathSciNet  Google Scholar 

  8. Melissen, J.B.M.: Optimal packings of eleven equal circles in an equilateral triangle. Acta Math. Hung. 65, 389–393 (1994)

    Article  MathSciNet  Google Scholar 

  9. Melissen, J.B.M., Schuur, P.C.: Packing 16, 17 or 18 circles on an equilateral triangle. Discret. Math. 145, 333–342 (1995)

    Article  MathSciNet  Google Scholar 

  10. Oler, N.: A finite packing problem. Can. Math. Bull. 4, 153–155 (1961)

    Article  MathSciNet  Google Scholar 

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  1. University of Dunaújváros, Dunaújváros, Hungary

    Antal Joós

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  1. Antal Joós
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Correspondence to Antal Joós.

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Supported by EFOP-3.6.1-16-2016-00003 funds, Consolidate long-term R and D and I processes at the University of Dunaujvaros.

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Joós, A. Packing 13 circles in an equilateral triangle. Aequat. Math. 95, 35–65 (2021). https://doi.org/10.1007/s00010-020-00753-y

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