Say you want to pack oranges in a box. How many can you fit?
Inevitably there will be some empty space, since spheres do not fit perfectly together. But after some experimenting, you might find that some arrangements are better than others.
Mathematicians consider an abstraction of this question called the sphere packing problem: how densely can we pack identical spheres in space without overlap?
The problem has occupied mathematicians for centuries. In 1611, Johannes Kepler conjectured that the greatest possible density in three dimensions was attained by the face-centred cubic (fcc) structure. While this was long believed to be true, it was only proved in 1998 by Thomas Hales (with extensive computer assistance).
The intuition for the Kepler conjecture can be shown by the following construction of the fcc structure (see figure), which shows the relationship with the densest one and two dimensional sphere packings.
In the one-dimensional sphere packing problem, the ‘spheres’ are simply line segments. By simply placing them end to end we can cover the whole line (one-dimensional space). The density of a packing is the proportion of space occupied by the spheres: here we can achieve the maximum density of 1.
To obtain the best two-dimensional packing, we first take the optimal one-dimensional arrangement and replace the line segments with circles, the two-dimensional ‘sphere’ analogue. Then we stack these rows as closely as possible, each circle occupying a ‘gap’ between circles in the previous row. This structure is called the hexagonal lattice, and it has a density of π/√12 ≈ 0.91.
Similarly, we can get dense three-dimensional packings by stacking layers of spheres in a hexagonal lattice arrangement. There are two ways to stack one hexagonal lattice on another; it’s not possible to place spheres in all the gaps of the layer below. Since packings are composed of infinitely many of these layers, there are an uncountably infinite number of them having density π/√18 ≈ 0.74. The fcc structure is the unique packing obtained by this method where the centres of the spheres form a lattice – a periodic array of points with a good algebraic structure.
Lattice packings are very important: they also provide some of the densest packings in higher dimensions. In 2016, Maryna Viazovska proved that the packing associated with the lattice E8 was the densest in eight dimensions. Soon afterward, she and her collaborators adapted the ideas to the 24-dimensional problem; the solution here is given by the Leech lattice. The E8 and Leech lattices have connections with many interesting areas of mathematics, such as ‘monstrous moonshine’.
As part of my project I studied root lattices, examples of which are the hexagonal, fcc, and E8 latttices. In particular, I looked at how they could be used to construct certain 24 dimensional lattices (Niemeier lattices) related to the Leech lattice.
Leo Jiang was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.