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]
| Number of circles |
Triangle number |
Length | Area | Figure |
|---|---|---|---|---|
| 1 | Yes | [math]\displaystyle{ 2 \sqrt {3} }[/math] = 3.464... | 5.196... | |
| 2 | [math]\displaystyle{ 2 + 2 \sqrt {3} }[/math] = 5.464... | 12.928... | ||
| 3 | Yes | [math]\displaystyle{ 2 + 2 \sqrt {3} }[/math] = 5.464... | 12.928... | |
| 4 | [math]\displaystyle{ 4 \sqrt {3} }[/math] = 6.928... | 20.784... | 
| |
| 5 | [math]\displaystyle{ 4 + 2 \sqrt {3} }[/math] = 7.464... | 24.124... | 
| |
| 6 | Yes | [math]\displaystyle{ 4 + 2 \sqrt {3} }[/math] = 7.464... | 24.124... | |
| 7 | [math]\displaystyle{ 2 + 4 \sqrt {3} }[/math] = 8.928... | 34.516... | ||
| 8 | [math]\displaystyle{ 2 + 2 \sqrt{3} + \tfrac {2} {3} \sqrt{33} }[/math] = 9.293... | 37.401... | ||
| 9 | [math]\displaystyle{ 6 + 2 \sqrt {3} }[/math] = 9.464... | 38.784... | ||
| 10 | Yes | [math]\displaystyle{ 6 + 2 \sqrt {3} }[/math] = 9.464... | 38.784... | |
| 11 | [math]\displaystyle{ 4 + 2 \sqrt {3} + \tfrac {4} {3} \sqrt{6} }[/math] = 10.730... | 49.854... | ||
| 12 | [math]\displaystyle{ 4 + 4 \sqrt {3} }[/math] = 10.928... | 51.712... | ||
| 13 | [math]\displaystyle{ 4 + \tfrac {10} {3} \sqrt{3} + \tfrac {2} {3} \sqrt{6} }[/math] = 11.406... | 56.338... | ||
| 14 | [math]\displaystyle{ 8 + 2 \sqrt {3} }[/math] = 11.464... | 56.908... | ||
| 15 | Yes | [math]\displaystyle{ 8 + 2 \sqrt {3} }[/math] = 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
- Circle packing in an isosceles right triangle
- Malfatti circles, three circles of possibly unequal sizes packed into a triangle
References
- ↑ 1.0 1.1 Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", The American Mathematical Monthly 100 (10): 916–925, doi:10.2307/2324212.
- ↑ 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, https://research.utwente.nl/en/publications/packing-16-17-of-18-circles-in-an-equilateral-triangle(b2172f19-9654-4ff1-9af4-59da1b6bef3d).html.
- ↑ "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), 1995, http://www.combinatorics.org/Volume_2/Abstracts/v2i1a1.html.
- ↑ Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin 4 (2): 153–155, doi:10.4153/CMB-1961-018-7.
- ↑ 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.
- ↑ 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, http://projecteuclid.org/getRecord?id=euclid.em/1045952348.
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