Sphere packing 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 $k=\lfloor p^{3}/2\rfloor$ , 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]

References

^ ^a ^b ^c Gensane, Th. (2004). "Dense packings of equal spheres in a cube". Electronic Journal of Combinatorics. 11 (1) Research Paper 33. doi:10.37236/1786. MR 2056085.
^ 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. MR 0200797.
^ Coxeter, MR200797
^ Schaer, J. (1994). "The densest packing of ten congruent spheres in a cube". Intuitive geometry (Szeged, 1991). Colloq. Math. Soc. János Bolyai. Vol. 63. Amsterdam: North-Holland. pp. 403–424. ISBN 0-444-81906-1. MR 1383635.
^ 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. MR 2504734.
^ Goldberg, Michael (1971). "On the densest packing of equal spheres in a cube". Mathematics Magazine. 44: 199–208. doi:10.2307/2689076. JSTOR 2689076. MR 0298562.
^ Tatarevic, Milos (2015). "On limits of dense packing of equal spheres in a cube". Electronic Journal of Combinatorics. 22 (1) Paper 1.35. arXiv:1503.07933. doi:10.37236/3784. MR 3315477.

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[gensane-1] Gensane, Th. (2004). "Dense packings of equal spheres in a cube". Electronic Journal of Combinatorics. 11 (1) Research Paper 33. doi:10.37236/1786. MR 2056085.

[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. MR 0200797.

[3] Coxeter, MR200797

[4] Schaer, J. (1994). "The densest packing of ten congruent spheres in a cube". Intuitive geometry (Szeged, 1991). Colloq. Math. Soc. János Bolyai. Vol. 63. Amsterdam: North-Holland. pp. 403–424. ISBN 0-444-81906-1. MR 1383635.

[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. MR 2504734.

[6] Goldberg, Michael (1971). "On the densest packing of equal spheres in a cube". Mathematics Magazine. 44: 199–208. doi:10.2307/2689076. JSTOR 2689076. MR 0298562.

[7] Tatarevic, Milos (2015). "On limits of dense packing of equal spheres in a cube". Electronic Journal of Combinatorics. 22 (1) Paper 1.35. arXiv:1503.07933. doi:10.37236/3784. MR 3315477.

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Sphere packing in a cube

See also

References