Image:Cannonball stack with FCC unit cell.jpg

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Shown above is what the science of sphere packing calls a closest-packed arrangement. Specifically, this is a modified form of the cannonball stack wherein three extra spheres have been added in order to help illustrate the organization of the FCC lattice.

Thomas Harriot in ca. 1585 first pondered the mathematics of cannonball stacks and later asked Johannes Kepler if the stack illustrated here was truly the most efficient. Kepler wrote, in what today is known as the Kepler conjecture, that no other arrangement of spheres can exceed its packing density of 74%.^[1]

Mathematically, there is an infinite quantity of closest-packed arrangements (assuming an infinite-size volume in which to arrange spheres). In the field of crystal structure however, unit cells (a crystal’s repeating pattern) are composed of a limited number of atoms and this reduces the variety of closest-packed regular lattices found in nature to only two: hexagonal close packed (HCP), and face-centered cubic (FCC). As can be seen at this site at King’s College, there is a distinct, real difference between different lattices; it’s not just a matter of how one slices 3D space. With all closest-packed lattices however, any given internal atom is in contact with 12 neighbors — the maximum possible. The cannonball stack shown here (which takes the form of a regular tetrahedron) has a face-centered cubic lattice.

Note that neither this stack nor the copper-colored subset constitute the FCC unit cell since neither can be tessellated in 3D space. Visit the Web site at King’s College to see FCC and HCP unit cells.

The stack shown here is indeed quite dense. If this stack of 35 spheres was composed of iron cannonballs, each measuring 10 cm in diameter, the top of the stack would be only 42.66 cm off the ground — just under the knee of the average barefoot man — and yet would weigh over 144 kg.

1. ^ To 23 significant digits, the value is 74.048 048 969 306 104 116 931%

Rendered and modeled using Ashlar Incorporated’s Cobalt on a Mac.

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