Circle packing in a circle explained

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 equivalent solution exists, all are shown.

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

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.)

See also

• Disk covering problem
• Square packing in a circle

References

1. Ekanayake . Dinesh . LaFountain . Douglas . Tight partitions for packing circles in a circle . Italian Journal of Pure and Applied Mathematics . 51 . 115–136.

^{[2] [3] [4] [5] [6] [7] [8]}

External links

• Mathematical analysis of 2D packing of circles (2022). H C Rajpoot from arXiv
• "The best known packings of equal circles in a circle (complete up to N = 2600)"
• "Online calculator for "How many circles can you get in order to minimize the waste?"