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]

A closely related problem is to cover the equilateral triangle with a given number of 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. ^ ^{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, MR1252928 .
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, MR1356610 .
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), MR1309122, http://www.combinatorics.org/Volume_2/Abstracts/v2i1a1.html .
4. ^ Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin 4: 153–155, doi:10.4153/CMB-1961-018-7, MR0133065 .
5. ^ Payan, Charles (1997), "Empilement de cercles égaux dans un triangle équilatéral. À propos d'une conjecture d'Erdős-Oler" (in French), Discrete Mathematics 165/166: 555–565, doi:10.1016/S0012-365X(96)00201-4, MR1439300 .
6. ^ Nurmela, Kari J. (2000), "Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles", Experimental Mathematics 9 (2): 241–250, MR1780209, http://projecteuclid.org/getRecord?id=euclid.em/1045952348 .