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.
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. This conjecture is now known to be true for n ≤ 15.
Minimum solutions for the side length of the triangle:
| 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.
See also
- Circle packing in an isosceles right triangle
- Malfatti circles, three circles of possibly unequal sizes packed into a triangle
References
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. /wiki/The_American_Mathematical_Monthly ↩
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. https://research.utwente.nl/en/publications/packing-16-17-of-18-circles-in-an-equilateral-triangle(b2172f19-9654-4ff1-9af4-59da1b6bef3d).html ↩
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. /wiki/Ronald_Graham ↩
Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065. /wiki/Canadian_Mathematical_Bulletin ↩
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. /wiki/Discrete_Mathematics_(journal) ↩
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. /wiki/The_American_Mathematical_Monthly ↩
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. http://projecteuclid.org/getRecord?id=euclid.em/1045952348 ↩