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 equivalent solution exists, all are shown.[1]
Number of unit circles |
Enclosing circle radius | Density | Optimality | Diagram |
---|---|---|---|---|
1 | 1 | 1.0000 | Trivially optimal. | |
2 | 2 | 0.5000 | 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?"