Sphere packing in a cube

{{Short description|Packing problem}}

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{{cite journal

| last = Gensane | first = Th.

| doi = 10.37236/1786

| issue = 1

| journal = Electronic Journal of Combinatorics

| mr = 2056085

| article-number = Research Paper 33

| title = Dense packings of equal spheres in a cube

| volume = 11

| year = 2004| doi-access = free

}} traces the origin of the problem to work of J. Schaer in the mid-1960s.{{cite journal

| last = Schaer | first = J.

| doi = 10.4153/CMB-1966-033-0

| journal = Canadian Mathematical Bulletin

| mr = 200797

| pages = 265–270

| title = On the densest packing of spheres in a cube

| volume = 9

| year = 1966}} 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".Coxeter, {{MR|200797}} The cases $k=7$ and $k=10$ were resolved in later work of Schaer,{{cite conference

| last = Schaer | first = J.

| contribution = The densest packing of ten congruent spheres in a cube

| isbn = 0-444-81906-1

| mr = 1383635

| pages = 403–424

| publisher = North-Holland | location = Amsterdam

| series = Colloq. Math. Soc. János Bolyai

| title = Intuitive geometry (Szeged, 1991)

| volume = 63

| year = 1994}} and a packing for $k=14$ was proven optimal by Joós.{{cite journal

| last = Joós | first = Antal

| doi = 10.1007/s10711-008-9308-3

| journal = Geometriae Dedicata

| mr = 2504734

| pages = 49–80

| title = On the packing of fourteen congruent spheres in a cube

| volume = 140

| year = 2009}} For larger numbers of spheres, all results so far are conjectural. In a 1971 paper, Goldberg found many non-optimal packings for other values of $k$ and three that may still be optimal.{{cite journal

| last = Goldberg | first = Michael

| doi = 10.2307/2689076

| journal = Mathematics Magazine

| jstor = 2689076

| mr = 298562

| pages = 199–208

| title = On the densest packing of equal spheres in a cube

| volume = 44

| year = 1971}} Gensane improved the rest of Goldberg's packings and found good packings for up to 32 spheres.

Goldberg also conjectured that for numbers of spheres of the form $k=\backslash lfloor\; p^3/2\backslash rfloor$, 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.{{cite journal

| last = Tatarevic | first = Milos

| doi = 10.37236/3784

| issue = 1

| journal = Electronic Journal of Combinatorics

| mr = 3315477

| article-number = Paper 1.35

| title = On limits of dense packing of equal spheres in a cube

| volume = 22

| year = 2015| arxiv = 1503.07933

}}