Circle packing in an equilateral triangle

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 2 3 {\displaystyle 2{\sqrt {3}}} = 3.464... 5.196...
2 2 + 2 3 {\displaystyle 2+2{\sqrt {3}}} = 5.464... 12.928...
3 Yes 2 + 2 3 {\displaystyle 2+2{\sqrt {3}}} = 5.464... 12.928...
4 4 3 {\displaystyle 4{\sqrt {3}}} = 6.928... 20.784...
5 4 + 2 3 {\displaystyle 4+2{\sqrt {3}}} = 7.464... 24.124...
6 Yes 4 + 2 3 {\displaystyle 4+2{\sqrt {3}}} = 7.464... 24.124...
7 2 + 4 3 {\displaystyle 2+4{\sqrt {3}}} = 8.928... 34.516...
8 2 + 2 3 + 2 3 33 {\displaystyle 2+2{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {33}}} = 9.293... 37.401...
9 6 + 2 3 {\displaystyle 6+2{\sqrt {3}}} = 9.464... 38.784...
10 Yes 6 + 2 3 {\displaystyle 6+2{\sqrt {3}}} = 9.464... 38.784...
11 4 + 2 3 + 4 3 6 {\displaystyle 4+2{\sqrt {3}}+{\tfrac {4}{3}}{\sqrt {6}}} = 10.730... 49.854...
12 4 + 4 3 {\displaystyle 4+4{\sqrt {3}}} = 10.928... 51.712...
13 4 + 10 3 3 + 2 3 6 {\displaystyle 4+{\tfrac {10}{3}}{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {6}}} = 11.406... 56.338...
14 8 + 2 3 {\displaystyle 8+2{\sqrt {3}}} = 11.464... 56.908...
15 Yes 8 + 2 3 {\displaystyle 8+2{\sqrt {3}}} = 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]

See also

References

  1. ^ a b Melissen, 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.
  5. ^ 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.
  6. ^ 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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