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
Triangle number
Length
Area
Figure
1
Yes
= 3.464...
5.196...
2
= 5.464...
12.928...
3
Yes
= 5.464...
12.928...
4
= 6.928...
20.784...
5
= 7.464...
24.124...
6
Yes
= 7.464...
24.124...
7
= 8.928...
34.516...
8
= 9.293...
37.401...
9
= 9.464...
38.784...
10
Yes
= 9.464...
38.784...
11
= 10.730...
49.854...
12
= 10.928...
51.712...
13
= 11.406...
56.338...
14
= 11.464...
56.908...
15
Yes
= 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]
Malfatti circles, three circles of possibly unequal sizes packed into a triangle
Referencesedit
^ abMelissen, 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.
^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.
^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.
^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.
^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.
circle, packing, equilateral, triangle, packing, problem, discrete, mathematics, where, objective, pack, unit, circles, into, smallest, possible, equilateral, triangle, optimal, solutions, known, triangular, number, circles, conjectures, available, conjecture,. 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 lt 13 and for any triangular number of circles and conjectures are available for n lt 28 1 2 3 A conjecture of Paul Erdos 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 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 6 See also editCircle packing in an isosceles right triangle Malfatti circles three circles of possibly unequal sizes packed into a triangleReferences edit 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 JSTOR 2324212 MR 1252928 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 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 Oler Norman 1961 A finite packing problem Canadian Mathematical Bulletin 4 2 153 155 doi 10 4153 CMB 1961 018 7 MR 0133065 Payan Charles 1997 Empilement de cercles egaux dans un triangle equilateral A propos d une conjecture d Erdos Oler Discrete Mathematics in French 165 166 555 565 doi 10 1016 S0012 365X 96 00201 4 MR 1439300 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 nbsp This elementary geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Circle packing in an equilateral triangle amp oldid 1198782341, wikipedia, wiki, book, books, library,
