Discriminant of the Icosahedral Group - Greg Egan

Discriminant of the Icosahedral Group

This image, created by Greg Egan, shows the ‘discriminant’ of the symmetry group of the icosahedron. This group acts as linear transformations of \(\mathbb{R}^3\) and thus also \(\mathbb{C}^3\). By a theorem of Chevalley, the space of orbits of this group action is again isomorphic to \(\mathbb{C}^3\). Each point in the surface shown here corresponds to a ‘nongeneric’ orbit: an orbit with fewer than the maximal number of points. More precisely, the space of nongeneric orbits forms a complex surface in \(\mathbb{C}^3\), called the discriminant, whose intersection with \(\mathbb{R}^3\) is shown here.