Geodesic
The Densest Packing of Five Spheres in a Cube
Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 271-274

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The purpose of this paper is to locate five points Pi ( l ≤ i ≤ 5 ) in a closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance i j between Pi and Pj. We prove that this minimum distance cannot exceed (=m, say), and if it is equal to m, then the corresponding configuration is congruent to the set of points shown in fig. 1, namely P1 = A1 (0,0,0), P2 = A8 (1, 1, 1), P3 = B1 (0,1/2,1), P4 = B3 (1/2,1,0) and P5 = B5 (1, 0, 1/2). 
Schaer, J. The Densest Packing of Five Spheres in a Cube. Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 271-274. doi: 10.4153/CMB-1966-034-8
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     title = {The {Densest} {Packing} of {Five} {Spheres} in a {Cube}},
     journal = {Canadian mathematical bulletin},
     pages = {271--274},
     year = {1966},
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     number = {3},
     doi = {10.4153/CMB-1966-034-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-034-8/}
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