Experimental evidence for the occurrence of E8 in nature and the radii of the Gosset circles

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a Zamolodchikov model involving the exceptional Lie group $E_8$. The model predicts 8 particles and predicts the ratio of their masses. I.e., the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of $E_8$. Under the 2-D (Peter McMullen) projection of the polytope, the image of the vertices are arranged in 8 concentric circles, here referred to as the Gosset circles. The Gosset circles are understood to correspond to the 8 masses in the model, and it is understood that the ratio of their radii is the same as the ratio of the corres-ponding conjectural masses. A ratio of the two smallest circles (read 2 smallest masses) is the golden number. The conjectures have been now validated experimentally, at least for the first five masses. The McMullen projection generalizes to any complex simple Lie algebra whose rank is greater than 1. The Gosset circles also generalize, using orbits of the Coxeter element. Using results in a 1959 paper of mine, I found some time ago a very easily defined operator $A$ whose spectrum is exactly the squares of the radii $r_i$ of these generalized Gosset circles. As a confirmation, in the $E_8$ case, using only the eigenvalues of a suitable multiple of $A$, Vogan computed the ratio of the $r_i$. Happily these agree with the corresponding ratio of the Zamolodchikov masses. The operator $A$ is written as a sum of $\ell +1$ rank 1 operators, parameterized by the points in the extended Dynkin diagram. Involved in this expansion are the coefficients $n_i$ of the highest root. Recalling the McKay correspondence, in the $E_8$ case, the $n_i$, together with 1, are the dimensions of the irreducible representations of the binary icosahedral group.