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 are what anyone would have guessed". [3] The cases and were resolved in later work of Schaer, [4] and a packing for 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 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 , 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
- ^ 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, MR 200797
- ^ 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.
 | This geometry-related article is a stub. You can help Wikipedia by expanding it. |
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 are what anyone would have guessed". [3] The cases and were resolved in later work of Schaer, [4] and a packing for 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 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 , 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
- ^ 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, MR 200797
- ^ 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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