Reminds me of a Kepler story, on this day of Fields math prizes.....
| |
Sir Walter Raleigh was an all-round Elizabethan scallywag. He wanted to know if there was a quick way of estimating the number of cannonballs in a pile.
| |
|
In 1606 this problem came to Johannes Kepler's attention, who considered how to pack them in the most efficient way.
Kepler experimented with different ways of stacking spheres. He concluded that the "face-centred cubic lattice" was the most efficient.
If you arrange 100 oranges (cannonballs were replaced by oranges for convenience's sake) in a flat layer of 10x10 and then place a similar layer directly on top, you have created a "simple cubic lattice". Provided your oranges haven't rolled apart, your pile has a packing efficiency of only 52% - you're effectively stacking as much air as oranges.
With Kepler's "face-centred cubic lattice" the first layer of oranges is formed in the same way you would spread penny coins on a desk to cover it leaving the least amount of gaps. Then for the second layer, place your fruit in the "dimples" created by the honeycomb beneath. Each successive layer is then built in the same way so the pile forms a pyramid. Using this method, Kepler calculated that the packing efficiency rose to 74%, constituting the highest efficiency you could ever get.
Or is it?
In 1998, Thomas Hales and his research student Samuel P. Fergusons devised an equation which modeled a cluster of 50 spheres. The equation and its 150 variables expressed every conceivable arrangement of these spheres. They then used computers to confirm that no combination of variables led to a packing efficiency higher than 74%. 250 pages of argument and 3 gigabytes of computer files proved Kepler right.