math – update

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Day #18: Reaching into Kepler´s Conjecture and Hilbert´s 18th Problem

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Q: Imagine a cube of sides L and put 8 – maximal – spheres centered on each vertex. Then calculate the density of this 8/8 spheres inside the cube!!… Now, put a whole sphere in the center of the cube – as dense as it´s possible – and calculate again!!… And now, take away the central sphere, and instead, put central spheres in each of the six faces of the cube – as dense as possible – and calculate again!!… Is this the maximal density of Kepler´s conjecture??…

The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.

In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales’ proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees have said that they are “99% certain” of the correctness of Hales’ proof, and now Kepler conjecture is accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants.


Diagrams of cubic close packing (left) and hexagonal close packing (right).

Imagine filling a large container with small equal-sized spheres. The density of the arrangement is equal to the collective volume of the spheres divided by the volume of the container. To maximize the number of spheres in the container means to create an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.

Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of

  {\frac  {\pi }{3{\sqrt  {2}}}}=0.740480489\ldots

The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.

After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of 23 unsolved problems of mathematics—it forms part of Hilbert’s 18th problem.

Source:´s Angle

kubisch-flächenzentrierte Packung links, hexagonale Packung rechts

Author: Math - Update

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