Two-state trajectory

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

Minimum solutions (in case several minimal solutions have been shown to exist, only one variant appears in the table):^[1]

Number of unit circles	Enclosing circle radius	Density	Optimality	Diagram
1	1	1.0000	Trivially optimal.
2	2	0.5000	Trivially optimal.
3	$1 + \frac{2}{3} \sqrt{3}$ ≈ 2.154...	0.6466...	Trivially optimal.
4	$1 + \sqrt{2}$ ≈ 2.414...	0.6864...	Trivially optimal.
5	$1 + \sqrt{2 (1 + \frac{1}{\sqrt{5}})}$ ≈ 2.701...	0.6854...	Proved optimal by Graham in 1968.^[2]	File:Disk pack5.svg
6	3	0.6667...	Proved optimal by Graham in 1968.^[2]	File:Disk pack6.svg
7	3	0.7778...	Proved optimal by Graham in 1968.^[2]	File:Disk pack7.svg
8	$1 + \frac{1}{\sin (\frac{π}{7})}$ ≈ 3.304...	0.7328...	Proved optimal by Pirl in 1969.^[3]	File:Disk pack8.svg
9	$1 + \sqrt{2 (2 + \sqrt{2})}$ ≈ 3.613...	0.6895...	Proved optimal by Pirl in 1969.^[3]	File:Disk pack9.svg
10	3.813...	0.6878...	Proved optimal by Pirl in 1969.^[3]	File:Disk pack10.svg
11	$1 + \frac{1}{\sin (\frac{π}{9})}$ ≈ 3.923...	0.7148...	Proved optimal by Melissen in 1994.^[4]	File:Disk pack11.svg
12	4.029...	0.7392...	Proved optimal by Fodor in 2000.^[5]	File:Disk pack12.svg
13	$2 + \sqrt{5}$ ≈4.236...	0.7245...	Proved optimal by Fodor in 2003.^[6]	File:Disk pack13.svg File:Disk pack13b.svg
14	4.328...	0.7474...	Conjectured optimal.^[7]	File:Disk pack14.svg
15	4.521...	0.7339...	Conjectured optimal.^[7]	File:Disk pack15.svg
16	4.615...	0.7512...	Conjectured optimal.^[7]	File:Disk pack16.svg
17	4.792...	0.7403...	Conjectured optimal.^[7]	File:Disk pack17.svg
18	$1 + \sqrt{2} + \sqrt{6}$ ≈ 4.863...	0.7611...	Conjectured optimal.^[7]	File:Disk pack18.svg
19	$1 + \sqrt{2} + \sqrt{6}$ ≈ 4.863...	0.8034...	Proved optimal by Fodor in 1999.^[8]	File:Disk pack19.svg
20	5.122...	0.7623...	Conjectured optimal.^[7]	File:Disk pack20.svg

References

↑ Erich Friedman, Circles in Circles on Erich's Packing Center
↑ ^2.0 ^2.1 ^2.2 R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.
↑ ^3.0 ^3.1 ^3.2 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.
↑ ^7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 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.

External links

"The best known packings of equal circles in a circle (complete up to N = 1500)" and application for "How many circles can you get in order to minimize the waste?"

Template:Packing problem

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[1] Erich Friedman, Circles in Circles on Erich's Packing Center

[Graham-2] 2.0 ^2.1 ^2.2 R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.

[Pirl-3] 3.0 ^3.1 ^3.2 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.

[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.

[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.

[Graham98-7] 7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 Graham RL, Lubachevsky BD, Nurmela KJ,Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

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

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Two-state trajectory

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Two-state trajectory

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