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	Is triangular	Length	Area
1	True	${\displaystyle 2{\sqrt {3}}}$ = 3.464...	5.196...
2	False	${\displaystyle 2+2{\sqrt {3}}}$ = 5.464...	12.928...
3	True	${\displaystyle 2+2{\sqrt {3}}}$ = 5.464...	12.928...
4	False	${\displaystyle 4{\sqrt {3}}}$ = 6.928...	20.784...
5	False	${\displaystyle 4+2{\sqrt {3}}}$ = 7.464...	24.124...
6	True	${\displaystyle 4+2{\sqrt {3}}}$ = 7.464...	24.124...
7	False	${\displaystyle 2+4{\sqrt {3}}}$ = 8.928...	34.516...
8	False	${\displaystyle 2+2{\sqrt {3}}+{\dfrac {2}{3}}{\sqrt {33}}}$ = 9.293...	37.401...
9	False	${\displaystyle 6+2{\sqrt {3}}}$ = 9.464...	38.784...
10	True	${\displaystyle 6+2{\sqrt {3}}}$ = 9.464...	38.784...
11	False	${\displaystyle 4+2{\sqrt {3}}+{\dfrac {4}{3}}{\sqrt {6}}}$ = 10.730...	49.854...
12	False	${\displaystyle 4+4{\sqrt {3}}}$ = 10.928...	51.712...
13	False	${\displaystyle 4+{\dfrac {10}{3}}{\sqrt {3}}+{\dfrac {2}{3}}{\sqrt {6}}}$ = 11.406...	56.338...
14	False	${\displaystyle 8+2{\sqrt {3}}}$ = 11.464...	56.908...
15	True	${\displaystyle 8+2{\sqrt {3}}}$ = 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

• Circle packing in an isosceles right triangle
• Malfatti circles, a construction giving the optimal solution for three circles in an equilateral triangle

References

1. 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.