Circle packing in a circle

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Short description: Two-dimensional packing problem

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.[1]

Number of
unit circles 		Enclosing circle radius Density Optimality Diagram
1 1 1.0000 Trivially optimal. Disk pack1.svg
2 2 0.5000 Trivially optimal. Disk pack2.svg
3 [math]\displaystyle{ 1+\frac{2}{\sqrt{3}} }[/math] ≈ 2.154... 0.6466... Trivially optimal. Disk pack3.svg
4 [math]\displaystyle{ 1+\sqrt{2} }[/math] ≈ 2.414... 0.6864... Trivially optimal. Disk pack4.svg
5 [math]\displaystyle{ 1+\sqrt{2\left(1+\frac{1}{\sqrt{5}}\right)} }[/math] ≈ 2.701... 0.6854... Proved optimal by Graham
(1968)[2] 		Disk pack5.svg
6 3 0.6666... Proved optimal by Graham
(1968)[2] 		Disk pack6.svg Disk pack6 2.svg
7 3 0.7777... Trivially optimal. Disk pack7.svg
8 [math]\displaystyle{ 1+\frac{1}{\sin\left(\frac{\pi}{7}\right)} }[/math] ≈ 3.304... 0.7328... Proved optimal by Pirl
(1969)[3] 		Disk pack8.svg
9 [math]\displaystyle{ 1+\sqrt{2\left(2+\sqrt{2}\right)} }[/math] ≈ 3.613... 0.6895... Proved optimal by Pirl
(1969)[3] 		Disk pack9.svg
10 3.813... 0.6878... Proved optimal by Pirl
(1969)[3] 		Disk pack10.svg
11 [math]\displaystyle{ 1+\frac{1}{\sin\left(\frac{\pi}{9}\right)} }[/math] ≈ 3.923... 0.7148... Proved optimal by Melissen
(1994)[4] 		Disk pack11.svg Disk pack11 2.svg
12 4.029... 0.7392... Proved optimal by Fodor
(2000)[5] 		Disk pack12.svg
13 [math]\displaystyle{ 2 + \sqrt{5} }[/math] ≈ 4.236... 0.7245... Proved optimal by Fodor
(2003)[6] 		Disk pack13.svg Disk pack13b.svg
14 4.328... 0.7474... Conjectured optimal by Goldberg
(1971).[7] 		Disk pack14.svg
15 [math]\displaystyle{ 1 + \sqrt{6 + \frac{2}{\sqrt{5}} + 4 \sqrt{1 +\frac{2}{\sqrt{5}}}} }[/math] ≈ 4.521... 0.7339... Conjectured optimal by Pirl
(1969).[7] 		Disk pack15.svg
16 4.615... 0.7512... Conjectured optimal by Goldberg
(1971).[7] 		Disk pack16.svg
17 4.792... 0.7403... Conjectured optimal by Reis
(1975).[7] 		Disk pack17.svg
18 [math]\displaystyle{ 1+\sqrt{2}+\sqrt{6} }[/math] ≈ 4.863... 0.7609... Conjectured optimal by Pirl (1969),
with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård (1998).[7] 		Disk pack18.svg 90px
90px 90px
90px 90px
90px 90px
90px Disk pack18 7.svg
19 [math]\displaystyle{ 1+\sqrt{2}+\sqrt{6} }[/math] ≈ 4.863... 0.8032... Proved optimal by Fodor
(1999)[8] 		Disk pack19.svg
20 5.122... 0.7623... Conjectured optimal by Goldberg (1971).[7] Disk pack20.svg

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, 19
  • Conjectured for n = 14, 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.)[9]

See also

References

  1. ↑ Friedman, Erich, "Circles in Circles", Erich's Packing Center, http://www2.stetson.edu/~efriedma/cirincir/ 
  2. ↑ 2.0 2.1 R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.
  3. ↑ 3.0 3.1 3.2 U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969) 111-124.
  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.
  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.
  9. ↑ Sloane, N. J. A., ed. "Sequence A084644". OEIS Foundation. https://oeis.org/A084644. 

External links





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