Universal Optimality of the Hexagonal Lattice

Symmetries and periodic structures are natural and fascinating and their appearances are manifold. Mathematics is capable of generating models of physical, chemical or biological processes, so that we can understand them. Certain structures may yield an equilibrium in a mathematical model, which is a satisfying answer to why objects arrange themselves as they do. One example is the crystal structure of salt, where sodium and chloride ions position themselves alternatingly on edges of cubes. A simpler model is given by a set of electrons which repel each other, without any other effects. In a very simple case we have an infinite collection of electrons, which must align on a straight line. The mathematical model provides the answer that the electrons will be arranged equidistantly. It may seem odd that, to arrange an infinite number of electrons on a line of infinite length is considered mathematically simple. Mathematics can deal very well with infinity. If we had a fixed number of electrons and limited space, then we would have to find conditions for the boundary and this yields a mathematically more complex model. An important feature of such models is, which mathematical function to choose for the electrostatic repulsion. The further away, the less the repulsion. But is it decreasing proportionally to the distance, or to the distance squared, or maybe exponentially to the distance? For each model we have to expect a different solution. Therefore we find a wealth of different structures in nature. But for our simple model, we always obtain the same regular pattern, nothing random or something like alternating short and long distances. The solution is said to be universally optimal. Universal optimality is a rare feature and thus highly important. Only recently, a mathematical incomparable breakthrough was achieved. The expected universally optimal structures in dimension 8 and 24 have been verified. On the other hand, it is known that in dimension 3 no universally optimal structure can exist. The current state of research tells us that there may provably exist a universally optimal structure in dimension 2: the hexagonal lattice. Each particle has 6 neighbors which form a regular hexagon. The new research approach is to use a certain sampling theorem to prove the universal optimality of the hexagonal lattice. Sampling theorems are mathematical statements telling us how many values of a signal (function) we need to know in order to know the entire signal. They have applications, e.g., in wireless communication (WLAN, 4G/5G technology) where they help to transmit data in an efficient and stable way. At first glance, the suggested method may seem inappropriate to solve the universal optimality conjecture in dimension 2. However, the problem of universal optima lity can be formulated as a sampling problem, which means that we have new mathematical tools at hand.

The project dealt with mathematical questions on discrete geometries in the plane. A simple way to formulate the question is the following. How should one distribute light sources in the plane such that the darkest point has the highest illumination? The main issue is that finding the darkest point is mathematically extremely difficult to describe and depends on many parameters, such as the number of sources we have per area or how we choose the arrangement. We found some geometric tricks and algebraic simplifications along the way, which allowed us to mathematically determine the best configuration with only grasping the minimum of the expected best configuration. The solution is that, no matter how many light sources we have per area, the hexagonal lattice (each source has six neighbors, forming a regular hexagon) is the best arrangement. This does not come as a big surprise as all numerical simulations suggested that this is the best possible way. However, up to our work this problem was mathematically completely open and our way of solving it, without knowing the darkest point in general, was a major breakthrough in pure mathematics. We showed that among an infinite number of configurations there is a single best option, whereas numerics may always only deal with a relatively small (finite) number of cases. The mathematical set-up is of such universal nature, that our findings are applicable in a variety of situations. In mathematical chemistry and physics our result belongs to the area of crystallization and gives mathematical explanation of certain ionic structures appearing in nature. In the field of data science and signal processing our results show that, for a certain given standard, there is a unique single sampling pattern which allows for the most stable transmission of data (such as speech) in wireless communication. Our results are a gate opener for future research in this direction in higher dimensional spaces. This may seem to be an artificial problem, but is for example of importance in enzyme encoding or data science.