Keplers Conjecture

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Kepler's Sphere-Packing Conjecture Is Finally Proved

One of his more obscure contributions to posterity however, lies in mathematics.

Raleigh wanted to know if there was a quick way of estimating the number of cannonballs in a pile. In this problem was presented to German astronomer, Johannes Kepler, who took it on but adapted it significantly.

His concern wasn't with how many cannonballs, but with 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 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".

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. Nature seems to dictate that a penny, surrounded by 6 others in a honeycomb arrangement, is best.

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Replicate this with your oranges. 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. But, how to prove it?

There are untold ways that oranges can be stacked and any of these might yield a higher percentage. The conjecture dogged mathematicians for centuries.

The 400-year-old Kepler Conjecture that your local greengrocer knows how to solve

The general feeling was that a proof might never appear. Then in , American Professor Thomas Hales stunned the world of mathematics. Aided by his research student Samuel P. Ferguson, Hales devised a monster equation based on a cluster of 50 spheres.

Submission history

In , 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. In , the formal proof was accepted by the journal Forum of Mathematics, Pi. 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. 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. The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density. He had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in Harriot was a friend and assistant of Sir Walter Raleigh , who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in , and went on to develop an early version of atomic theory.

This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one.

Mathematical mysteries: Kepler's conjecture |

But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume is now known to always reduce their density. After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century.

In David Hilbert included it in his list of twenty three unsolved problems of mathematics —it forms part of Hilbert's eighteenth problem. This meant that a proof by exhaustion was, in principle, possible. Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. In , Wu-Yi Hsiang claimed to have proven the Kepler conjecture. The current consensus is that Hsiang's proof is incomplete.

Kepler conjecture

In , assisted by his graduate student Samuel Ferguson, he embarked on a research program to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5, different configurations of spheres. If a lower bound for the function value could be found for every one of these configurations that was greater than the value of the function for the cubic close packing arrangement, then the Kepler conjecture would be proved.

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To find lower bounds for all cases involved solving about , linear programming problems. When presenting the progress of his project in , Hales said that the end was in sight, but it might take "a year or two" to complete. In August Hales announced that the proof was complete.

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