Sphere packing in a sphere

Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions.

class="wikitable" ! rowspan=2 \| Number of inner spheres ! colspan=2 \| Maximum radius of inner spheres[http://oeis.org/A084829 Best packing of m>1 equal spheres in a sphere setting a new density record] ! rowspan=2 \| Packing density ! rowspan=2 \| Optimality ! rowspan=2 \| Arrangement ! rowspan=2 \| Diagram
Exact form ! Approximate
align=center \| 1 \| $1$ \| 1.0000 \| 1 \| Trivially optimal. \| Point \| 120px
align=center \| 2 \| $\dfrac {1} {2}$ \| 0.5000 \| 0.25 \| Trivially optimal. \| Line segment \| 120px
align=center \| 3 \| $2 \sqrt {3} - 3$ \| 0.4641... \| 0.29988... \| Trivially optimal. \| Triangle \| 120px
align=center \| 4 \| $\sqrt {6} - 2$ \| 0.4494... \| 0.36326... \| Proven optimal. \| Tetrahedron \| 120px
align=center \| 5 \| $\sqrt {2} - 1$ \| 0.4142... \| 0.35533... \| Proven optimal. \| Trigonal bipyramid \| 120px
align=center \| 6 \| $\sqrt {2} - 1$ \| 0.4142... \| 0.42640... \| Proven optimal. \| Octahedron \| 120px
align=center \| 7 \| $\frac {1}{\frac {\sqrt {3} + 2 \cos \left( \frac {\pi}{18} \right)}{\sqrt {2 + 2 \sqrt {3} \cos \left( \frac {\pi}{18} \right)}} + 1}$ \| 0.3859... \| 0.40231... \| Proven optimal. \| Capped octahedron \| 120px
align=center \| 8 \| $\frac {1}{\sqrt {2 + \frac {1}{\sqrt {2}}} + 1}$ \| 0.3780... \| 0.43217... \| Proven optimal. \| Square antiprism \| 120px
align=center \| 9 \| $\frac {\sqrt {3} - 1}{2}$ \| 0.3660... \| 0.44134... \| Proven optimal. \| Tricapped trigonal prism \| 120px
align=center \| 10 \| \| 0.3530... \| 0.44005... \| Proven optimal. \| \| 120px
align=center \| 11 \| $\dfrac {\sqrt{5} - 3} {2} + \sqrt{5 - 2 \sqrt{5} }$ \| 0.3445... \| 0.45003... \| Proven optimal. \| Diminished icosahedron \| 120px
align=center \| 12 \| $\dfrac {\sqrt{5} - 3} {2} + \sqrt{5 - 2 \sqrt{5} }$ \| 0.3445... \| 0.49095... \| Proven optimal. \| Icosahedron \| 120px

class="wikitable"

! rowspan=2 | Number of
inner spheres

! colspan=2 | Maximum radius of inner spheres[http://oeis.org/A084829 Best packing of m>1 equal spheres in a sphere setting a new density record]

! rowspan=2 | Packing
density

! rowspan=2 | Optimality

! rowspan=2 | Arrangement

! rowspan=2 | Diagram

Exact form

! Approximate

align=center

| 1

| $1$