Circle packing in an equilateral triangle

Unsolved problem in mathematics

What is the smallest possible equilateral triangle which an amount

n

of unit circles can be packed into?

More unsolved problems in mathematics

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 have been proved for $n \leq 15$ , and for any triangular number of circles, and conjectures are available for $n \leq 34$ .^[1]^[2]^[3]^[4]

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.^[5] This conjecture is now known to be true for $n \leq 15$ .^[6] In a paper by Graham and Lubachevsky concerning solutions for $22 \leq n \leq 34$ they also conjectured seven infinite families of optimal solutions in addition to the one by Erdős and Oler. These families give conjectured solutions for many more numbers including $n = 37, 40, 42, 43, 46, 49$ ^[3]

Minimum solutions for the side length of the triangle:^[1]

Number of circles	Triangle number	Length	Area
1	Yes	$2{\sqrt {3}}$ = 3.464...	5.196...
2		$2+2{\sqrt {3}}$ = 5.464...	12.928...
3	Yes	$2+2{\sqrt {3}}$ = 5.464...	12.928...
4		$4{\sqrt {3}}$ = 6.928...	20.784...
5		$4+2{\sqrt {3}}$ = 7.464...	24.124...
6	Yes	$4+2{\sqrt {3}}$ = 7.464...	24.124...
7		$2+4{\sqrt {3}}$ = 8.928...	34.516...
8		$2+2{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {33}}$ = 9.293...	37.401...
9		$6+2{\sqrt {3}}$ = 9.464...	38.784...
10	Yes	$6+2{\sqrt {3}}$ = 9.464...	38.784...
11		$4+2{\sqrt {3}}+{\tfrac {4}{3}}{\sqrt {6}}$ = 10.730...	49.854...
12		$4+4{\sqrt {3}}$ = 10.928...	51.712...
13		$4+{\tfrac {10}{3}}{\sqrt {3}}+{\tfrac {2}{3}}{\sqrt {6}}$ = 11.406...	56.338...
14		$8+2{\sqrt {3}}$ = 11.464...	56.908...
15	Yes	$8+2{\sqrt {3}}$ = 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.^[7]

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, 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.
^ ^a ^b 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.
^ Tedeschi, Natalie "On Packing Thirteen Points in an Equilateral Triangle", Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, https://doi.org/10.33697/ajur.2021.042
^ 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 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.

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[Melissen-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, JSTOR 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.

[:0-3] 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] Tedeschi, Natalie "On Packing Thirteen Points in an Equilateral Triangle", Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, https://doi.org/10.33697/ajur.2021.042

[5] Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.

[6] 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.

[7] 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.

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v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Doyle spiral Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma

Circle packing in an equilateral triangle

Circle packing in an equilateral triangle

Circle packing in an equilateral triangle

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