A214813 Maximal contact number of a subset of n balls from the face-centered cubic lattice.

0, 1, 3, 6, 9, 12, 15, 18, 21

Offset: 1

Author

N. J. A. Sloane, Jul 31 2012

Keywords

• nonn
• more

Comments

If S is an arrangement of non-overlapping balls of radius 1, the contact number of S is the number of pairs of balls that just touch each other.

a(13) >= 36 (take one ball and its 12 neighbors), so this is different from A008486.

If b(n) denotes the maximal contact number of any arrangement of n balls then it is conjectured that a(n) = b(n) for n <= 9. It is also known that b(10)>=25, b(11)>=29, b(12)>=33 and of course b(13) >= a(13) >= 36. [Bezdek 2012]

Note that Figure 1e of Bezdek's arxiv:1601.00145 shows at n=5 a sphere packing with 9 contacts on the hexagonal close package (!), not on the cubic close package (which equals the f.c.c.). [In Figure 1e there is one sphere that touches from above a set of 3 spheres in a middle layer right above the bottom sphere; so this needs the ABABA... layer structures of the h.c.p, and cannot be done with the ABCABC... layer structure of the f.c.c.] So Figure 1e is not demonstrating a(5)=9. The correct value for the f.c.c is apparently a(5)=8 (where two structures with 8 contacts exist.) - R. J. Mathar, Mar 13 2018

Links

• Bezdek, Karoly, Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space, Discrete Comput. Geom. 48 (2012), no. 2, 298--309. MR2946449
• K. Bezdek, M. A. Khan, Contact number for sphere packings, arXiv:1601.00145 [math.MG], 2016.
• K. Bezdek, S. Reid, Contact graphs of unit sphere packings revisited, J. Geom. 104 (1) (2013) 57-83.
• J. P. K. Doye, D. J. Wales, Magic numbers and growth sequences of small face-centered-cubic and decahedral clusters, Chem. Phys. Lett. 247 (1995) 339, Table 1 column n(fcc).
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Index entries for sequences related to f.c.c. lattice

Crossrefs

Cf. A004015, A005901, A038173.