In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres and this conjecture was proven by T. C. Highest density is only in case of 1,2,3,8 and 24 dimensions. Many crystal structures are based on a close-packing of a kind of atom. The cubic and hexagonal arrangements are very close to one another in energy, there are two simple regular lattices that achieve this highest average density. They are called face-centered cubic and hexagonal close-packed, based on their symmetry, both are based upon sheets of spheres arranged at the vertices of a triangular tiling, they differ in how the sheets are stacked upon one another. The fcc lattice is known to mathematicians as that generated by the A3 root system. Cannonballs were usually piled in a rectangular or triangular wooden frame, both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a pyramid with a hexagonal base. In both the fcc and hcp arrangements each sphere has twelve neighbors, for every sphere there is one gap surrounded by six spheres and two smaller gaps surrounded by four spheres. The distances to the centers of these gaps from the centers of the spheres is √ 3⁄2 for the tetrahedral, and √2 for the octahedral. Relative to a layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the one is possible. The most regular ones are fcc = ABCABCA hcp = ABABABA, in close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch of one sphere diameter. The coordination number of hcp and fcc is 12 and their atomic packing factors are equal to the mentioned above,0.74. The distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere, in close packing all of the spheres share a common radius, r. Therefore two centers would simply have a distance 2r, hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres centers
Cannonballs piled on a triangular (front) and rectangular (back) base, both fcc lattices.
Snowballs stacked in preparation for a snowball fight. The front pyramid is hexagonal close-packed and rear is face-centered cubic.
Image: Hexagonal close packed unit cell
Image: Close packed spheres, with umbrella light & camerea