Packing 13 circles in an equilateral triangle
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Aequationes Mathematicae
Packing 13 circles in an equilateral triangle ´s Antal Joo
Abstract. The maximum separation problem is to find the maximum of the minimum pairwise distance of n points in a planar body B on the Euclidean plane. In this paper this problem will be considered if 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. Mathematics Subject Classification. 52C15. Keywords. Packing, Circle, Equilateral triangle, Separation distance.
1. Introduction It is a classical problem to determine the smallest circle, the smallest square or, alternatively, the smallest equilateral triangle that can contain n congruent nonoverlapping circles [1–4]. The problem of finding the largest common radius rn of n congruent circles that can be packed inside the unilateral triangle is equivalent to maximizing the minimum pairwise distance dn of n points in the unilateral triangle. It is easily seen that there is a homothety between the unilateral triangle enclosing the circles and the smallest equilateral triangle containing their centers. In the sequel, the latter problem will be called the maximum separation problem and dn denotes the maximum separation distance of n points in the unilateral triangle. Table 1 shows the best known results up to 19. Thus the optimal dn are known for n ≤ 12 and n = k(k + 1)/2 if k is an integer. Graham and Lubachevsky [5] presented lower bounds for n = 22, . . . , 34. Supported by EFOP-3.6.1-16-2016-00003 funds, Consolidate long-term R and D and I processes at the University of Dunaujvaros.
´s A. Joo
AEM
Table 1. The maximum separation distances up to 19 n
dn
Approximation
Source
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 k(k + 1)/2
1 1 √ 1/ 3 1/2 1/2 √ (√3 − 1)/2 ( 33 − 3)/8 1/3 1/3 √ (3 −√ 6)/2 2− 3
=1 =1 = 0.577350 . . . = 0.5 = 0.5 = 0.366025 . . . = 0.343070 . . . = 0.333333 . . . = 0.333333 . . . = 0.275255 . . . = 0.267949 . . . ≥ 0.251813 . . . = 0.25 = 0.25 ≥ 0.216227 . . . = 0.211324 . . . ≥ 0.203465 . . . ≥ 0.200321 . . .
Elementary Elementary [7] [7] [6,10] [7] [7] [7] [6,10] [8] [7] [5,7] [7] [6,10] [5,9] [5,7,9] [5,9] [5,7] [6,10]
≥ 1/4 1/4 ≥ (3 −
√
3)/6
1/(k − 1)
The aim of this paper is to consider the case n = 13. Melissen [7] gave the arrangement that can be seen in Fig. 1 and 0.251813 . . . was the first approximation of this separation distance. Melissen conjectured that d13 = 0.251813 . . .. Graham and Lubachevsky [5] approximated this separation distance to 0.251813236653061 and conjectured the same. In this paper we will calculate the exact separation distance of the points in Fig. 1. Moreover we will prove the above conjecture of Melissen and the above conjecture of Graham and Lubachevsky.
2. The result Theorem 1. The maximum separation distance of 13 points in the equilateral triangle of side length 1 is √ √ √ 7 6
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