Sphere packing in a cube

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Short description: Packing problem
Spheres packed in a cube

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=⌊p3/2⌋, 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

References

  1. ↑ 1.0 1.1 1.2 Gensane, Th. (2004). "Dense packings of equal spheres in a cube". Electronic Journal of Combinatorics 11 (1). doi:10.37236/1786. 
  2. ↑ Schaer, J. (1966). "On the densest packing of spheres in a cube". Canadian Mathematical Bulletin 9: 265–270. doi:10.4153/CMB-1966-033-0. 
  3. ↑ Coxeter, MR200797
  4. ↑ Schaer, J. (1994). "Intuitive geometry (Szeged, 1991)". 63. Amsterdam: North-Holland. pp. 403–424. ISBN 0-444-81906-1. 
  5. ↑ Joós, Antal (2009). "On the packing of fourteen congruent spheres in a cube". Geometriae Dedicata 140: 49–80. doi:10.1007/s10711-008-9308-3. 
  6. ↑ Goldberg, Michael (1971). "On the densest packing of equal spheres in a cube". Mathematics Magazine 44: 199–208. doi:10.2307/2689076. 
  7. ↑ Tatarevic, Milos (2015). "On limits of dense packing of equal spheres in a cube". Electronic Journal of Combinatorics 22 (1). doi:10.37236/3784. 




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