Random thoughts on densest packing

I really liked Paul Chaikin’s Reference Frame, “Random Thoughts” (Physics Today, June 2007, page 8 ). However, he overlooks the problem of uniqueness. He states that face-centered cubic packing “has recently been proven to be the densest packing.” However, FCC can’t be the densest because hexagonal close packing is just as dense. FCC and HCP have exactly the same packing density, ¹ 0.74. That something with a certain property exists doesn’t automatically make it unique with respect to that property. Furthermore, both FCC and HCP have equal thermodynamic stability according to the ideal gas laws that Chaikin presents.

The lack of uniqueness has a certain relevance to the issues of random ordering. Suppose a random ordering is found that has more thermodynamic stability than a crystal. Other forms of random ordering may have the same degree of thermodynamic stability. However, can one even define a random order that is also unique?