High-dimensional spheres can be stacked chaotically particularly well
The problem of the densest sphere packing sounds simple, but is surprisingly difficult to crack. Now four young mathematicians have achieved the first breakthrough in 75 years.
For hundreds of years, experts have been working on how to arrange spheres in the most space-saving way. The problem apparently goes back to the navigator Sir Walter Raleigh, who travelled the world's oceans in the 16th century. He commissioned his scientific advisor at the time, Thomas Harriot, with a seemingly simple task: Harriot was to calculate the minimum amount of floor space required to stack a certain number of cannonballs in a pyramid shape. Unbeknownst to him at the time, Raleigh's task would shape the coming centuries of mathematics. Numerous puzzles are still associated with it today.
Although Harriot was able to quickly solve the problem set by Raleigh, questions arose: What is the most space-efficient arrangement if you want to fill the entire three-dimensional space with an infinite number of spheres? This problem, known as Kepler's conjecture, was not solved until 1998 with the help of computers. Since then, experts have succeeded in calculating the densest packing for two-dimensional discs in the plane as well as for three-dimensional spheres in space (Kepler's problem) and even in higher dimensions. For example, there is a clear answer to the question of how 8- and 24-dimensional "spheres" can be arranged in a corresponding 8- or 24-dimensional space to save as much space as possible. However, the general result for d-dimensional spheres is still uncertain.
But there are estimates. Until recently, the best estimate came from mathematician Claude Ambrose Rogers, who proved in 1947 that the density of d-dimensional spheres must be at least 2⁄e-d-2-d. This estimate remained valid in the following decades - it could hardly be improved.
The improvement may seem modest, but the problem is very difficult and the method is new.
However, the four mathematicians Marcelo Campos, Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe have now been able to increase Roger's estimate by a factor of log(d), as they report in a paper that has not yet been peer-reviewed and was published in December 2023. This is the first major advance in the field in 75 years. "This improvement may seem modest, but the problem is known to be very difficult and the method used is new," writes mathematician Timothy Gowers on X.
The chaos as a solution
In order to make such estimates, possible arrangements of spheres are usually analysed and their density, i.e. the proportion of spheres in space, is calculated. It is very complicated to prove that this is really the densest packing: You have to prove that every other arrangement - ordered or disordered - always has a lower sphere density.
Estimating the density is a little easier. You can start with the simplest possible arrangement: Suppose you want to pack an infinite number of d-dimensional spheres with a radius of one into a d-dimensional space. To do this, you first pick out the set of all points in the space that have a distance of two from each other. If you place a sphere with a radius of one at each of these points, you have found a possible arrangement - albeit not the optimal one. This has a density of 2-d. So the densest sphere packing must have a density of at least 2-d.
Rogers discovered that this estimate can be improved - by a factor of 2⁄e-d. Essentially, he showed that there is an arrangement of d-dimensional spheres that have this density. In the following decades, other experts were only able to improve Roger's result by a constant factor. "It is striking that the papers use quite different methods and yet all lead to the same linear improvement of the trivial bound," wrote Fields medallist Akshay Venkatesh, for example.
Campos, Jenssen, Michelen and Sahasrabudhe have now used stochastic methods in their work to investigate new arrangements of spheres. What is astonishing is that their result is not based on an ordered arrangement, but on chaotically distributed spheres. Sometimes chaos leads to the goal - even in mathematics.
Spectrum of Science
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