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.

Number of inner spheres	Maximum radius of inner spheres^[1]		Packing density	Optimality	Arrangement	Diagram
Number of inner spheres	Exact form	Approximate	Packing density	Optimality	Arrangement	Diagram
1	$1$	1.0000	1	Trivially optimal.	Point
2	${\dfrac {1}{2}}$	0.5000	0.25	Trivially optimal.	Line segment
3	$2{\sqrt {3}}-3$	0.4641...	0.29988...	Trivially optimal.	Triangle
4	${\sqrt {6}}-2$	0.4494...	0.36326...	Proven optimal.	Tetrahedron
5	${\sqrt {2}}-1$	0.4142...	0.35533...	Proven optimal.	Trigonal bipyramid
6	${\sqrt {2}}-1$	0.4142...	0.42640...	Proven optimal.	Octahedron
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
8	${\frac {1}{{\sqrt {2+{\frac {1}{\sqrt {2}}}}}+1}}$	0.3780...	0.43217...	Proven optimal.	Square antiprism
9	${\frac {{\sqrt {3}}-1}{2}}$	0.3660...	0.44134...	Proven optimal.	Tricapped trigonal prism
10		0.3530...	0.44005...	Proven optimal.
11	${\dfrac {{\sqrt {5}}-3}{2}}+{\sqrt {5-2{\sqrt {5}}}}$	0.3445...	0.45003...	Proven optimal.	Diminished icosahedron
12	${\dfrac {{\sqrt {5}}-3}{2}}+{\sqrt {5-2{\sqrt {5}}}}$	0.3445...	0.49095...	Proven optimal.	Icosahedron

References[edit]

^ Best packing of m>1 equal spheres in a sphere setting a new density record

[1] Best packing of m>1 equal spheres in a sphere setting a new density record

[1]

v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma

Sphere packing in a sphere

References[edit]

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