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<p><b>New page</b></p><div>'''Circle packing in a circle''' is a two-dimensional [[packing problem]] with the objective of packing unit circles into the smallest possible larger [[circle]].<br />
<br />
Minimum solutions (in case several minimal solutions have been shown to exist, only one variant appears in the table):<ref>[http://www2.stetson.edu/~efriedma/cirincir/ Erich Friedman, ''Circles in Circles'' on Erich's Packing Center]</ref><br />
<br />
{| class="wikitable"<br />
! Number of<br>unit circles<br />
! Enclosing<br>circle radius<br />
! Density<br />
! Optimality<br />
! Diagram<br />
|- align=center<br />
| 1<br />
| 1<br />
| 1.0000<br />
| Trivially optimal.<br />
| [[Image:Disk pack1.svg|120x120px]]<br />
|- align=center<br />
| 2<br />
| 2<br />
| 0.5000<br />
| Trivially optimal.<br />
| [[Image:Disk pack2.svg|120x120px]]<br />
|- align=center<br />
| 3<br />
| <math>1+\frac{2}{3} \sqrt{3}</math><br>≈ 2.154...<br />
| 0.6466...<br />
| Trivially optimal.<br />
| [[Image:Disk pack3.svg|120x120px]]<br />
|- align=center<br />
| 4<br />
| <math>1+\sqrt{2}</math><br>≈ 2.414...<br />
| 0.6864...<br />
| Trivially optimal.<br />
| [[Image:Disk pack4.svg|120x120px]]<br />
|- align=center<br />
| 5<br />
| <math>1+\sqrt{2(1+\frac{1}{\sqrt{5}})}</math><br>≈ 2.701...<br />
| 0.6854...<br />
| Proved optimal<br>by Graham in 1968.<ref name="Graham">R.L. Graham, ''Sets of points with given minimum separation (Solution to Problem El921)'', Amer. Math. Monthly 75 (1968) 192-193.</ref><br />
| [[Image:Disk pack5.svg|120x120px]]<br />
|- align=center<br />
| 6<br />
| 3<br />
| 0.6667...<br />
| Proved optimal<br>by Graham in 1968.<ref name="Graham"/><br />
| [[Image:Disk pack6.svg|120x120px]]<br />
|- align=center<br />
| 7<br />
| 3<br />
| 0.7778... <br />
| Proved optimal<br>by Graham in 1968.<ref name="Graham"/><br />
| [[Image:Disk pack7.svg|120x120px]]<br />
|- align=center<br />
| 8<br />
| <math>1+\frac{1}{\sin(\frac{\pi}{7})}</math><br>≈ 3.304...<br />
| 0.7328...<br />
| Proved optimal<br>by Pirl in 1969.<ref name="Pirl">U. Pirl, ''Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten'', ''[[Mathematische Nachrichten]]'' 40 (1969) 111-124.</ref><br />
| [[Image:Disk pack8.svg|120x120px]]<br />
|- align=center<br />
| 9<br />
| <math>1+\sqrt{2(2+\sqrt{2})}</math><br>≈ 3.613...<br />
| 0.6895...<br />
| Proved optimal<br>by Pirl in 1969.<ref name="Pirl"/><br />
| [[Image:Disk pack9.svg|120x120px]]<br />
|- align=center<br />
| 10<br />
| 3.813...<br />
| 0.6878...<br />
| Proved optimal<br>by Pirl in 1969.<ref name="Pirl"/><br />
| [[Image:Disk pack10.svg|120x120px]]<br />
|- align=center<br />
| 11<br />
| <math>1+\frac{1}{\sin(\frac{\pi}{9})}</math><br>≈ 3.923...<br />
| 0.7148...<br />
| Proved optimal<br>by Melissen in 1994.<ref name="Melissen">H. Melissen, ''Densest packing of eleven congruent circles in a circle'', ''[[Geometriae Dedicata]]'' 50 (1994) 15-25.</ref><br />
| [[Image:Disk pack11.svg|120x120px]]<br />
|- align=center<br />
| 12<br />
| 4.029...<br />
| 0.7392...<br />
| Proved optimal<br>by Fodor in 2000.<ref>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.</ref><br />
| [[File:Disk pack12.svg|120px]]<br />
|- align=center<br />
| 13<br />
| <math>2 + \sqrt{5}</math><br>≈4.236...<br />
| 0.7245...<br />
| Proved optimal<br>by Fodor in 2003.<ref>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.</ref><br />
| [[File:Disk pack13.svg|120x120px]] [[File:Disk pack13b.svg|120x120px]]<br />
|- align=center<br />
| 14<br />
| 4.328...<br />
| 0.7474...<br />
| Conjectured optimal.<ref name="Graham98">Graham RL, Lubachevsky BD, Nurmela KJ,Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.</ref><br />
| [[File:Disk pack14.svg|120px]]<br />
|- align=center<br />
| 15<br />
| 4.521...<br />
| 0.7339...<br />
| Conjectured optimal.<ref name="Graham98"/><br />
| [[File:Disk pack15.svg|120px]]<br />
|- align=center<br />
| 16<br />
| 4.615...<br />
| 0.7512...<br />
| Conjectured optimal.<ref name="Graham98"/><br />
| [[File:Disk pack16.svg|120px]]<br />
|- align=center<br />
| 17<br />
| 4.792...<br />
| 0.7403...<br />
| Conjectured optimal.<ref name="Graham98"/><br />
| [[File:Disk pack17.svg|120px]]<br />
|- align=center<br />
| 18<br />
| <math>1+\sqrt{2}+\sqrt{6}</math><br>≈ 4.863...<br />
| 0.7611...<br />
| Conjectured optimal.<ref name="Graham98"/><br />
| [[File:Disk pack18.svg|120px]]<br />
|- align=center<br />
| 19<br />
| <math>1+\sqrt{2}+\sqrt{6}</math><br>≈ 4.863...<br />
| 0.8034...<br />
| Proved optimal<br>by Fodor in 1999.<ref>F. Fodor, ''The Densest Packing of 19 Congruent Circles in a Circle'', Geom. Dedicata 74 (1999), 139–145.</ref><br />
| [[File:Disk pack19.svg|120px]]<br />
|- align=center<br />
| 20<br />
| 5.122...<br />
| 0.7623...<br />
| Conjectured optimal.<ref name="Graham98"/><br />
| [[File:Disk pack20.svg|120px]]<br />
|}<br />
<br />
==See also==<br />
*[[Disk covering problem]]<br />
<br />
==References==<br />
<references/><br />
<br />
== External links ==<br />
* [http://hydra.nat.uni-magdeburg.de/packing/cci "The best known packings of equal circles in a circle (complete up to N = 1500)" and application for "How many circles can you get in order to minimize the waste?"]<br />
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{{Packing problem}}<br />
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[[Category:Circle packing]]<br />
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{{elementary-geometry-stub}}</div>
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