Sphere packing in a cube explained

In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.

Gensane^[1] traces the origin of the problem to work of J. Schaer in the mid-1960s.^[2] Reviewing Schaer's work, H. S. M. Coxeter writes that he "proves that the arrangements for

k =2 , 3 , 4 , 8 , 9 are what anyone would have guessed".^[3] The cases k =7 and k =10 were resolved in later work of Schaer,^[4] and a packing for k =14 was proven optimal by Joós.^[5] For larger numbers of spheres, all results so far are conjectural.^[1] In a 1971 paper, Goldberg found many non-optimal packings for other values of k and three that may still be optimal.^[6] Gensane improved the rest of Goldberg's packings and found good packings for up to 32 spheres.^[1]

Goldberg also conjectured that for numbers of spheres of the form

k = \lfloor p ^{3/2\rfloor} , the optimal packing of spheres in a cube is a form of cubic close-packing. However, omitting as few as two spheres from this number allows a different and tighter packing.^[7]

See also

• Packing problem
• Sphere packing in a cylinder

Notes and References

1. Gensane . Th. . 10.37236/1786 . 1 . Electronic Journal of Combinatorics . 2056085 . Research Paper 33 . Dense packings of equal spheres in a cube . 11 . 2004. free .
2. Schaer . J. . 10.4153/CMB-1966-033-0 . Canadian Mathematical Bulletin . 200797 . 265–270 . On the densest packing of spheres in a cube . 9 . 1966.
3. Coxeter,
4. Schaer . J. . The densest packing of ten congruent spheres in a cube . 0-444-81906-1 . 1383635 . 403–424 . North-Holland . Amsterdam . Colloq. Math. Soc. János Bolyai . Intuitive geometry (Szeged, 1991) . 63 . 1994.
5. Joós . Antal . 10.1007/s10711-008-9308-3 . Geometriae Dedicata . 2504734 . 49–80 . On the packing of fourteen congruent spheres in a cube . 140 . 2009.
6. Goldberg . Michael . 10.2307/2689076 . Mathematics Magazine . 2689076 . 298562 . 199–208 . On the densest packing of equal spheres in a cube . 44 . 1971.
7. Tatarevic . Milos . 10.37236/3784 . 1 . Electronic Journal of Combinatorics . 3315477 . Paper 1.35 . On limits of dense packing of equal spheres in a cube . 22 . 2015. 1503.07933 .