Circle packing in an equilateral triangle information
Two-dimensional packing problem
Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number of circles, and conjectures are available for n < 28.[1][2][3]
A conjecture of Paul Erdős and Norman Oler states that, if n is a triangular number, then the optimal packings of n − 1 and of n circles have the same side length: that is, according to the conjecture, an optimal packing for n − 1 circles can be found by removing any single circle from the optimal hexagonal packing of n circles.[4] This conjecture is now known to be true for n ≤ 15.[5]
Minimum solutions for the side length of the triangle:[1]
Number of circles
Triangle number
Length
Area
Figure
1
Yes
= 3.464...
5.196...
2
= 5.464...
12.928...
3
Yes
= 5.464...
12.928...
4
= 6.928...
20.784...
5
= 7.464...
24.124...
6
Yes
= 7.464...
24.124...
7
= 8.928...
34.516...
8
= 9.293...
37.401...
9
= 9.464...
38.784...
10
Yes
= 9.464...
38.784...
11
= 10.730...
49.854...
12
= 10.928...
51.712...
13
= 11.406...
56.338...
14
= 11.464...
56.908...
15
Yes
= 11.464...
56.908...
A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.[6]
^ abMelissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", The American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928.
^Melissen, J. B. M.; Schuur, P. C. (1995), "Packing 16, 17 or 18 circles in an equilateral triangle", Discrete Mathematics, 145 (1–3): 333–342, doi:10.1016/0012-365X(95)90139-C, MR 1356610.
^Graham, R. L.; Lubachevsky, B. D. (1995), "Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond", Electronic Journal of Combinatorics, 2: Article 1, approx. 39 pp. (electronic), MR 1309122.
^Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.
^Payan, Charles (1997), "Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler", Discrete Mathematics (in French), 165/166: 555–565, doi:10.1016/S0012-365X(96)00201-4, MR 1439300.
^Nurmela, Kari J. (2000), "Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles", Experimental Mathematics, 9 (2): 241–250, doi:10.1080/10586458.2000.10504649, MR 1780209, S2CID 45127090.
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