# Endrass Octic

An octic surface is one defined by a polynomial equation of degree 8. The Endrass octic, drawn above by Abdelaziz Nait Merzouk, is currently the octic surface with the largest known number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$. It has 168 ordinary double points, while the best known upper bound for a octic surface that’s smooth except for such singularities is 174.

# Labs Septic

A septic surface is one defined by a polynomial equation of degree 7. The Labs septic, drawn above by Abdelaziz Nait Merzouk, is a septic 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 $x^2 + y^2 = z^2$.

# 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 $x^2 + y^2 = z^2$.

# Rectified Truncated Icosahedron

The rectified truncated icosahedron is a surprising new polyhedron discovered by Craig S. Kaplan. It has a total of 60 triangles, 12 pentagons and 20 hexagons as faces.

# Zamolodchikov Tetrahedron Equation

The Zamolodchikov tetrahedron equation, illustrated above by J. Scott Carter and Masahico Saito, is a fundamental law governing surfaces embedded in 4-dimensional space. It also arises purely algebraically in the theory of braided monoidal 2-categories.