# Circle packing in an equilateral triangle

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 Length
1 ${\displaystyle 2{\sqrt {3}}}$ = 3.464...
2 ${\displaystyle 2+2{\sqrt {3}}}$ = 5.464...
3 ${\displaystyle 2+2{\sqrt {3}}}$ = 5.464...
4 ${\displaystyle 4{\sqrt {3}}}$ = 6.928...
5 ${\displaystyle 4+2{\sqrt {3}}}$ = 7.464...
6 ${\displaystyle 4+2{\sqrt {3}}}$ = 7.464...
7 ${\displaystyle 2+4{\sqrt {3}}}$ = 8.928...
8 ${\displaystyle 2+2{\sqrt {3}}+{\dfrac {2}{3}}{\sqrt {33}}}$ = 9.293...
9 ${\displaystyle 6+2{\sqrt {3}}}$ = 9.464...
10 ${\displaystyle 6+2{\sqrt {3}}}$ = 9.464...
11 ${\displaystyle 4+2{\sqrt {3}}+{\dfrac {4}{3}}{\sqrt {6}}}$ = 10.730...
12 ${\displaystyle 4+4{\sqrt {3}}}$ = 10.928...
13 ${\displaystyle 4+{\dfrac {10}{3}}{\sqrt {3}}+{\dfrac {2}{3}}{\sqrt {6}}}$ = 11.406...
14 ${\displaystyle 8+2{\sqrt {3}}}$ = 11.464...
15 ${\displaystyle 8+2{\sqrt {3}}}$ = 11.464...

A closely related problem is to cover the equilateral triangle with a given number of circles, having as small a radius as possible.[6]

## 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, 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: 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.