Circle packing in a circle

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

Number of
unit circles		 Enclosing circle diameter Density Optimality Diagram
1 1 1.0000 Trivially optimal.
2 2 0.5000 Trivially optimal.
3 ≈ 2.154... 0.6466... Trivially optimal.
4 ≈ 2.414... 0.6864... Trivially optimal.
5 ≈ 2.701... 0.6854... Trivially optimal.
Also proved optimal by Graham
(1968)[2]
6 3 0.6666... Trivially optimal.
Also 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... Conjectured optimal.[7]
15 ≈ 4.521... 0.7339... Conjectured optimal.[7]
16 4.615... 0.7512... Conjectured optimal.[7]
17 4.792... 0.7403... Conjectured optimal.[7]
18 ≈ 4.863... 0.7611... Conjectured optimal.[7]
19 ≈ 4.863... 0.8034... Proved optimal by Fodor
(1999)[8]
20 5.122... 0.7623... Conjectured optimal.[7]

See also

References
  1. Friedman, Erich, "Circles in Circles", Erich's Packing Center, archived from the original on 2020-03-18
  2. R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.
  3. 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. 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.


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.