Packing Spheres

Is there another way of packing spheres that is more space-efficient?

In 1611 Johannes Kepler asserted that there was no way of packing equivalent spheres at a greater density than that of a face-centred cubic arrangement. This is now known as the Kepler Conjecture.



4 atoms in the unit cell (0, 0, 0) (0, 1 /2, 1 /2) (1 /2, 0, 1 /2) (1 /2, 1 /2, 0)








HCP, 2 atoms in the unit cell (0, 0, 0) (2/3, 1 /3, 1 /2)

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/Lec1.html#anchor5




How to stack oranges

..... Kepler settled on an arrangement known as the face-centred cubic, which also happens to be the way greengrocers stack oranges.

Using this arrangement, oranges occupy 74.04 per cent of the total space. Kepler could not find a more efficient way to stack spheres, but nor could he be sure that no such arrangement exists. With an infinite number of possible arrangements, the challenge has been to prove categorically whether Kepler's suggested arrangement is best.

Prof Hales's approach to the problem is based on a single equation with more than 150 variables, which can be changed to describe every conceivable arrangement, thereby allowing the equation to calculate the packing efficiency for each one. Traditionally, mathematicians would alter the variables to maximise the packing efficiency for the equation, and then see which arrangement is associated with the variables. However, the equation is hugely complex, which puts the maximisation process beyond paper and pencil calculations, and even challenges the limits of computers.

Over the past decade, Prof Hales, helped by his research student Samuel Ferguson, has been studying the maximisation process, inventing shortcuts which bring it within the realm of computability. At last, having thrown enough computer power at the problem and testing all possible arrangements, Prof Hales has concluded that no arrangement beats the face-centred cubic for efficiency. In other words, Kepler and greengrocers have been right all along.
Simon Singh,

http://www.chem.ox.ac.uk