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 optimal solution exists, all are shown.
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... 0.6461... Trivially optimal. 4 2.414... 0.6864... Trivially optimal. 5 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 3.304... 0.7328... Proved optimal by Pirl(1969) 9 3.613... 0.6895... Proved optimal by Pirl(1969) 10 3.813... 0.6878... Proved optimal by Pirl(1969) 11 3.923... 0.7148... Proved optimal by Melissen(1994) 12 4.029... 0.7392... Proved optimal by Fodor(2000) 13 4.236... 0.7245... Proved optimal by Fodor(2003) 14 4.328... 0.7474... Proved optimal by Ekanayake and LaFountain(2024). 15 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 4.863... 0.7609... Conjectured optimal by Pirl (1969),with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård (1998). 19 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 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. F. Fodor, The Densest Packing of 19 Congruent Circles in a Circle, Geom. Dedicata 74 (1999), 139–145. R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193. Graham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154. 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.
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?"