Description
Pick any integer N greater than one. Now I ask you to place N points on a sphere so that the points are as far apart as possible. Where should you place the points? If you have chosen N=2, then placing one point at the North Pole and one point at the South Pole will obviously be a correct solution. However, if N is greater than 2 then you quickly see that it is not even clear what I meant by as far apart as possible. The question needs to be more precise.
For example, I could ask that the smallest distance between any two points is as large as possible. This is known as the Tammes problem, named after the Dutch botanist who posed the problem in connection with the study of the arrangement of pores in spherical pollen grains.
Another way to get the points to be far apart from each other is to imagine that they are particles that repel each other, for example, they might be particles that all have the same electric charge. This is called the Thomson problem, named after the British physicist that discovered the electron and proposed the plum pudding model of the atom.
For some very small values of N both the Tammes and Thomson problems can be solved and have the same solution, for example, for N=4 the four points should be placed on the vertices of a regular tetrahedron. However, in general the two problems have different solutions and these can only be found by computer calculations. The image shows some solutions of the Thomson problem for N=4,12,32.
It turns out that the solutions to these (and similar) problems have applications in a range of topics from modelling viruses, to carbon chemistry, and designing golf balls.
The project is to understand and investigate the above problems by writing some computer codes in Python to find the solutions, visualize them, and to calculate some of their properties. For example, it turns out that their are some magic numbers for N at which the solution has a lot of symmetry.
Prerequisites
None.Resources
There are many webpages dedicated to these kinds of problems. Some examples are
The Thomson problem on Wikipedia
The Cambridge Cluster Database