It is well-known that through stereographic projection, one can put a complex coordinate z on a spherical surface. Felix Klein studied the complex coordinates of the vertices, edge centres and face centres of each platonic solid this way, and found that they are the roots of rather simple polynomials in z. Related to these Klein polynomials there are some further, rational functions of z (ratios of polynomials), which have the same symmetries as the platonic solids.
Recently, it has been discovered that various model physical systems, in chemistry, condensed matter, nuclear and particle physics, have smooth structures with the same symmetries as platonic solids. The Klein polynomials and related rational functions are very useful for describing them mathematically.
The talk will end with a discussion of a model for atomic nuclei in which the protons and neutrons are regarded as close enough together
to partially merge into one or other of these symmetric structures. Various small nuclei, up to carbon-12 and a bit larger, have been
modelled this way.
