- Circle packing in an equilateral triangle
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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 3.464... 2 5.464... 3 5.464... 4 6.928... 
5 7.464... 
6 7.464... 7 8.928... 8 9.293... 9 9.464... 10 9.464... 11 10.730... 12 10.928... 13 11.406... 14 11.464... 15 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]
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
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin 4: 153–155, doi:10.4153/CMB-1961-018-7, MR0133065.
- ^ 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.
- ^ 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.
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