Packing Spheres in High Dimensions

V eit Elser of Cornell University has just described a new way to elucidate a problem that has baffled mathematicians for over a century: How densely can identical spheres be packed as the dimension of the spheres and of the space they occupy grow ever larger [1]? Schemes for densely packing spheres have been worked out in low dimensions and for two special cases: 8 and 24 dimensions. (A sphere in n-dimensional space is a set of points that are the same fixed distance away from a given center point.) Surprisingly, some schemes in high dimensions are little better than a random approach. What’s more, a century of research has failed to improve on the result from Hermann Minkowski, a Germanmathematician who came up with the concept of four-dimensional spacetime, that with each additional dimension, the highest fraction of space that can be occupied by spheres falls by a factor of 2. Intuitively, the rate of decrease should be slower.