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.
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.)
See also
*
Disk covering problem The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point i ...
*
Square packing in a circle
Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle.
Square packing in a square
Square packing in a square is the problem of determining ...
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 Rajpootfrom
arXiv
"The best known packings of equal circles in a circle (complete up to N = 2600)"
Circle packing
{{elementary-geometry-stub