Circle packing in a circle

Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle.

Table of solutions, 1 ≤ n ≤ 20

If more than one optimal solution exists, all are shown.^[1]

$n$	Enclosing circle radius $r$	Density $n\!/r^{2}$	Optimality
1	1	1.0000...	Trivially optimal.
2	2	0.5000...	Trivially optimal.
3	2.155... $1+{\frac {2}{\sqrt {3}}}$	0.6466...	Trivially optimal.
4	2.414... $1+{\sqrt {2}}$	0.6864...	Trivially optimal.
5	2.701... $1+{\sqrt {2\left(1+{\frac {1}{\sqrt {5}}}\right)}}$	0.6854...	Proved optimal by Graham (1968)^[2]
6	3	0.6666...	Proved optimal by Graham (1968)^[2]
7	3	0.7777...	Trivially optimal.
8	3.304... $1+{\frac {1}{\sin {\frac {\pi }{7}}}}$	0.7328...	Proved optimal by Pirl (1969)^[3]
9	3.613... $1+{\sqrt {2\left(2+{\sqrt {2}}\right)}}$	0.6895...	Proved optimal by Pirl (1969)^[3]
10	3.813...	0.6878...	Proved optimal by Pirl (1969)^[3]
11	3.923... $1+{\frac {1}{\sin {\frac {\pi }{9}}}}$	0.7148...	Proved optimal by Melissen (1994)^[4]
12	4.029...	0.7392...	Proved optimal by Fodor (2000)^[5]
13	4.236... $2+{\sqrt {5}}$	0.7245...	Proved optimal by Fodor (2003)^[6]
14	4.328...	0.7474...	Proved optimal by Ekanayake and LaFountain (2024).^[7]
15	4.521... $1\!+\!{\sqrt {6\!+\!{\frac {2}{\sqrt {5}}}\!+\!4{\sqrt {1\!+\!{\frac {2}{\sqrt {5}}}}}}}$	0.7339...	Conjectured optimal by Pirl (1969).^[8]
16	4.615...	0.7512...	Conjectured optimal by Goldberg (1971).^[8]
17	4.792...	0.7403...	Conjectured optimal by Reis (1975).^[8]
18	4.863... $1+{\sqrt {2}}+{\sqrt {6}}$	0.7609...	Conjectured optimal by Pirl (1969), with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård (1998).^[8]
19	4.863... $1+{\sqrt {2}}+{\sqrt {6}}$	0.8032...	Proved optimal by Fodor (1999)^[9]
20	5.122...	0.7623...	Conjectured optimal by Goldberg (1971).^[8]

Special cases

Only 26 optimal packings are thought to be rigid (with no circles able to "rattle"). Numbers in bold are prime:

Proven for n = 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 19
Conjectured for n = 15, 16, 17, 18, 22, 23, 27, 30, 31, 33, 37, 61, 91

Of these, solutions for n = 2, 3, 4, 7, 19, and 37 achieve a packing density greater than any smaller number > 1. (Higher density records all have rattles.)^[10]

References

^ Friedman, Erich, "Circles in Circles", Erich's Packing Center, archived from the original on 2020-03-18
^ ^a ^b R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.
^ ^a ^b ^c U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969) 111-124.
^ H. Melissen, Densest packing of eleven congruent circles in a circle, Geometriae Dedicata 50 (1994) 15-25.
^ F. Fodor, The Densest Packing of 12 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 41 (2000) ?, 401–409.
^ F. Fodor, The Densest Packing of 13 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 44 (2003) 2, 431–440.
^ Ekanayake, Dinesh; LaFountain, Douglas. "Tight partitions for packing circles in a circle" (PDF). Italian Journal of Pure and Applied Mathematics. 51: 115–136.
^ ^a ^b ^c ^d ^e Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.
^ F. Fodor, The Densest Packing of 19 Congruent Circles in a Circle, Geom. Dedicata 74 (1999), 139–145.
^ Sloane, N. J. A. (ed.). "Sequence A084644". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

External links

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[:0-1] Friedman, Erich, "Circles in Circles", Erich's Packing Center, archived from the original on 2020-03-18

[Graham68-2] R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.

[Pirl-3] U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969) 111-124.

[Melissen-4] H. Melissen, Densest packing of eleven congruent circles in a circle, Geometriae Dedicata 50 (1994) 15-25.

[Fodor1-5] F. Fodor, The Densest Packing of 12 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 41 (2000) ?, 401–409.

[Fodor2-6] F. Fodor, The Densest Packing of 13 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 44 (2003) 2, 431–440.

[7] Ekanayake, Dinesh; LaFountain, Douglas. "Tight partitions for packing circles in a circle" (PDF). Italian Journal of Pure and Applied Mathematics. 51: 115–136.

[Graham98-8] Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

[Fodor3-9] F. Fodor, The Densest Packing of 19 Congruent Circles in a Circle, Geom. Dedicata 74 (1999), 139–145.

[10] Sloane, N. J. A. (ed.). "Sequence A084644". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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Circle packing in a circle

Contents

Table of solutions, 1 ≤ n ≤ 20

Special cases

See also

References

External links