# Factor for envelope circle for a group of equal circles

Solutions for the smallest diameter circles into which $n$ unit-diameter circles can be packed have been proved optimal for $n$ = 1 through 10, based on a paper by Kravitz, S. 'Packing Cylinders into Cylindrical Containers' Math. Mag. 40, 65-70, 1967.

Symbol
$F_{eq}$
Formulae
 $1$ $N_c<=1$ $2$ $N_c<=2$ $1+0.6667\sqrt{3}$ $N_c<=3$ $1+\sqrt{2}$ $N_c<=4$ $1+\sqrt{2\left(1+\frac{1}{\sqrt{5}}\right)}$ $N_c<=5$ $1+\frac{1}{sin\left(\frac{\pi}{6}\right)}$ $N_c<=7$ $1+\frac{1}{sin\left(\frac{\pi}{7}\right)}$ $N_c<=8$ $1+\sqrt{2\left(2+\sqrt{2}\right)}$ $N_c<=9$
Related
Used in
$D_{eq}$
$W_{conv,od}$
$W_{conv,og}$
Choices
NFactorEquationDensityOptimalityReference
111.0trivially optimal
220.5trivially optimal
32.154$1+2/\sqrt 3$0.6466trivially optimal
42.414$1+\sqrt 2$0.6864trivially optimal
52.701$1+\sqrt (2(1+1/\sqrt 5))$0.6854proved optimalGraham (1968)
63$1+1/sin(\pi/6)$0.6666proved optimalGraham (1968)
73$1+1/sin(\pi/6)$0.7777trivially optimal
83.304$1+1/sin(\pi/7)$0.7328proved optimal
93.613$1+\sqrt (2(2+\sqrt 2))$0.6895proved optimalPirl (1969)
103.8130.6878proved optimalPirl (1969)
113.923$1+1/sin(\pi/9)$0.7148proved optimalMelissen (1994)
124.0290.7392proved optimalFodor (2000)
134.236$2+\sqrt 5$0.7245proved optimalFodor (2003)
144.3280.7474conjectured optimal
154.521$1+\sqrt (6+2/\sqrt 2+4\sqrt (1+2/\sqrt 5))$0.7339conjectured optimal
164.6150.7512conjectured optimal
174.7920.7403conjectured optimal
184.863$1+\sqrt 2 +\sqrt 6$0.7611conjectured optimal
194.863$1+\sqrt 2 +\sqrt 6$0.8034proved optimalFodor (1999)
205.1220.7623conjectured optimal