The ends are trihedral ( i . e ., composed of three planes ) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120? the angle that minimizes surface area for a given volume . ( The angle formed by the edges at the pyramidal apex, known as the tetrahedral angle, is approximately 109?28'16 " ( arccos ( " 1 / 3 ) } } ) .)
42.
:: : : I'd bet that the best such packing for N = 120 is to inscribe the spheres in the dodecahedra of the 120-cell . & ndash; In S ^ 2, the surface of the 3-ball, when N > > 12 you typically get regions of mildly distorted " kissing pennies " lattice, with 12 flaws where a disk has 5 neighbors rather than 6; in other words, the Voronoi cells are usually 12 pentagons and N-12 hexagons, increasingly regular as N increases . & ndash; Since face-centred cubic is the densest packing of balls in flat 3-space, I'm guessing that it shows up in curved 3-space, mildly distorted, when N is big enough . & mdash; Tamfang 19 : 19, 21 September 2007 ( UTC)
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