Kepler’s conjecture

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

G.G. Szpiro, Kepler s Conjecture How Some of the Greatest Minds in History... 

For an interesting history of the Kepler conjecture, see G. G. Szpiro, Kepler s Conjecture How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World, Wiley, Hoboken, NJ, 2003. 

Recall Kepler s conjecture on the density of packing uniform spheres outlined in the opening section of this book. The proof by Hales [58] was based on an exhaustive check by a computer of a very large, but finite, number of different arranganents. For a finite number of possibilities, there is little conceptual distinction between an exhaustive check of all possibilities and use of the trial-and-error approach to a check of all possibilities. Both could be very inefficient, in contrast to the exact solutions, which do not involve searches, which make them fairly efficient. 

An obvious possible improvement of the Bohr model was to bring it better into line with Kepler s model of the solar sxstem, which placed the planets in elliptical, rather than circular, orbits. Sommerfeld managed to solve this problem by the introduction of two extra quantum numbers in addition to the principal quantum number (n) of the Bohr model, and the formulation of general quantization rules for periodic systems, which contained the Bohr conjecture as a special case. 

With the availability of computers, it has been recognized that not only can some problems (which up to that time were beyond the possibility of being numerically solved) be reexamined and solved, but also new problems can emerge, be discovered, considered, and even solved, which were unknown in pre-computer time. Let us briefly mention two such problems that have some novelty, both of which have some connection with chemistry (i) the proof or almost a proof of the Kepler conjecture about dense packing of spheres and (ii) the problan relating to Ulam s spiral. 

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