A133587 Conjectured order of the symmetry group of the (numerically computed) least-perimeter cluster of n nonoverlapping circles.

4, 6, 4, 2, 10, 12, 14, 2, 4, 2, 6, 2, 4, 1, 2, 2, 2, 12, 2, 2, 1

Offset: 2

Author

Keenan Pepper, Dec 27 2007

Keywords

• nonn

Comments

This can be thought of as the order of the symmetry group of the minimum-energy configuration of n two-dimensional bubbles in a plane. a(1) is infinite, because one bubble forms a circle, which has a continuous symmetry group containing rotations of arbitrary angles. So far, the actual symmetry groups are all dihedral, except for a(15) and a(22), which are trivial (their configurations have no symmetries).

Examples

			a(3) = 6 because three planar bubbles arrange themselves in an equilateral-triangle-type configuration with symmetry group D_3, of order 6.
		

References

• Cox, S. J., F. Graner, M. F. Vaz, C. Monnereau-Pittet and N. Pittet, 2003, Minimal perimeter for N identical bubbles in two dimensions: calculations and simulations, Philos. Mag. 83, 1393-1406.
• F. Morgan, Soap bubble clusters, Rev. Mod. Phys. Vol. 79 (2007), pp. 821-827.

Links

• R. L. Graham and N. J. A. Sloane, Penny-Packing and Two-Dimensional Codes, Discrete and Comput. Geom. 5 (1990), 1-11.

Crossrefs

Cf. A133491.