Circle packing in an equilateral triangle

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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... 4 cirkloj en 60 60 60 triangulo.png
5 7.464... 5 cirkloj en 60 60 60 triangulo v1.png 5 cirkloj en 60 60 60 triangulo v2.png
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[edit]

References[edit]

  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 .