A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

2, 3, 4, 6, 8, 9, 10, 11, 12, 20, 24, 32, 38, 42, 44, 48

Offset: 1

Author

Hugo Pfoertner, Apr 07 2021

Keywords

• nonn
• more

Comments

This sequence is analogous to A084829, sharing the terms up to 12, but with the restriction that only arrangements are considered in which all small spheres touch the surface of the container sphere and the inner area remains empty. Formal proofs of optimality exist only for arrangements up to and including 14 and for 24 spheres, but the last improvements in the range of the specified terms were found before 1994. For references and links see A080865.

A conjectured continuation after the term 48 is 72, 78, 84, 92, 98, 120.

The linked illustration also shows a fitted curve estimating the minimum density achievable by optimal solutions of the Tammes problem for large n. The fitted equation is rho_min(n) = 0.565854 - 1/(0.566242*n + 2.67822). For comparison, consider the highest attainable density of spheres arranged in a flat hexagonal grid. This density is 0.604599788... = Pi * sqrt(3)/9. Achieving this density is made more difficult in the curved surface layer of a sphere because with large n there must always be 12 neighborhoods where the spheres packed in this layer can only have 5 nearest neighbors.

Examples

			  a(n)  Volume fraction in layer (rounded)
   2    0.25000
   3    0.30000
   4    0.36364
   6    0.42857
   8    0.43853
   9    0.45000
  10    0.45152
  11    0.46397
  12    0.50615
  20    0.51162
  24    0.52941
  32    0.53205
  38    0.53373
  42    0.53439
  44    0.54286
  48    0.54993
		

Links

• Hugo Pfoertner, Illustration of volume fraction of spheres in surface layer, (2021).

Crossrefs

Cf. A080865, A084829, A121346.