Kepler conjecture

It is not hard to understand why many metals favor an fee crystal structure there is no packing of hard spheres in space that creates a higher density than the fee structure. (A mathematical proof of this fact, known as the Kepler conjecture, has only been discovered in the past few years.) There is, however, one other packing that has exactly the same density as the fee packing, namely the hexagonal close-packed (hep) structure. As our third example of applying DFT to a periodic crystal structure, we will now consider the hep metals. 

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. 

This problem relates to infinite networks of polyhedral forms and is related to topics of crystallography. We will discuss in one of the coming sections the Kepler conjecture, which stated that in three-dimensional Euclidean space the most dense packing of spheres of equal size is cubic and hexagonal close packing. As we will see, this Kepler conjecture has been more recently solved, but the solution is based on a computer-assisted proof, which is a novelty, considered by some even to be to some extent controversial because of its lack of verifiability by a human reader in a... 

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. 

The Kepler conjecture received attention from outstanding mathematicians in the past, including Gauss and Hilbert, and was only relatively recently solved. Thomas Hales [119] used computer to exhaustively check various possible, very large but finite, numbers of different arrangements (it was shown by the Hungarian mathematician Fejes T6th in 1953 that the problem can be so reduced [120]). The proof of Hales took... 

T. C. Hales, A compnter verification of the Kepler conjecture, in Proc. Int. Congress Mathematicians, Vol. II. Invited lectures. Held in Beijing, August 20-28, 2002, T. Li (Ed.), Higher Education Press, Beijing, China, 2002, pp. 795-804. 

Hales, T.C. (2005) A proof of the Kepler conjecture, Ann. Math. 162, 1065. The paper is 121 pages long Twelve reviewers spent more than 4 years reviewing it. 

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... 

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. 

The second section deals with the fine structure of the hydrogen spectral lines. The discussion of the bound-state Kepler problem is extended to the relativistic case. Following an idea of Bohr [1], who had already conjectured that the fine structure of the hydrogen spectrum could be a relativistic effect proportional to the fine structure of the hydrogen spectrum is... 

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. 

It is impossible to pack spheres of equal size with a higher relative density than in the fee and hep structure. This has been conjectured by Johannes Kepler in 1611, but it was proven only in 1999 by Hales und Ferguson, using the power of modern computer algebra [136]. 

It is almost ironic that physics, at its most advanced level, has recently taught us that some of these conjectures, which even motivated Kepler to speculate about the harmony of our cosmos, have an amazing parallel in our... 

Because the metallic bond is nondirectional and because the positive ion cores attract each other through the intervening sea of electrons that flows between them, metals tend to form close-packed structures, meaning they try to get as dose together as they can in order to maximize their coordination nmnber. Johaimes Kepler (a mathematician as well as the famous astronomer) conjectured that the maximum number of identical spheres that could surround and touch another sphere was 12. Now, let us see what kinds of structures are possible with a coordination number of 12. 

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