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
editIf more than one optimal solution exists, all are shown.[1]
| Enclosing circle radius |
Density |
Optimality | Layout(s) of the circles | |
|---|---|---|---|---|
| 1 | 1 | 1.0 | Trivially optimal. | 
|
| 2 | 2 | 0.5 | Trivially optimal. | 
|
| 3 | 2.155... |
0.6466... | Trivially optimal. | 
|
| 4 | 2.414... |
0.6864... | Trivially optimal. | 
|
| 5 | 2.701... |
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... |
0.7328... | Proved optimal by Pirl (1969)[3] |
|
| 9 | 3.613... |
0.6895... | Proved optimal by Pirl (1969)[3] |
|
| 10 | 3.813... | 0.6878... | Proved optimal by Pirl (1969)[3] |
|
| 11 | 3.923... |
0.7148... | Proved optimal by Melissen (1994)[4] |
 
|
| 12 | 4.029... | 0.7392... | Proved optimal by Fodor (2000)[5] |
|
| 13 | 4.236... |
0.7245... | Proved optimal by Fodor (2003)[6] |
 
|
| 14 | 4.328... | 0.7474... | Proved optimal by Ekanayake and LaFountain (2024).[7] |
|
| 15 | 4.521... |
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... |
0.7609... | Conjectured optimal by Pirl (1969), with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård (1998).[8] |
         
|
| 19 | 4.863... |
0.8032... | Proved optimal by Fodor (1999)[9] |
|
| 20 | 5.122... | 0.7623... | Conjectured optimal by Goldberg (1971).[8] | 
|
Special cases
editOnly 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]
See also
editReferences
edit- ^ 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
edit- 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?"
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