# 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.

# Barth Sextic

A sextic surface is one defined by a polynomial equation of degree 6. The Barth sextic, drawn above by Craig Kaplan, is the sextic surface with the maximum possible number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$.

# Clebsch Surface

This is an image of the Clebsch surface, created by Greg Egan. The Clebsch surface owes its fame to the fact that while all smooth cubic surfaces defined over the complex numbers contain 27 lines, for this particular example all the lines are real, and thus visible to the eye. However, it has other nice properties as well.

# Branched Cover from (4 4 3/2) Schwarz Triangle

A Schwarz triangle is a spherical triangle that can be used to generate a tiling of a branched covering of the sphere by repeatedly reflecting this triangle across its edges. Sometimes we get an actual tiling of the sphere, but in general we get a branched covering, because the same point can lie in the interior of several triangles, and there may be branch points at the corners of the triangles.