# Laves Graph

This picture by Greg Egan shows the Laves graph, a structure discovered by the crystallographer Fritz Laves in 1932. It is also called the ‘$\mathrm{K}_4$ crystal’, since is an embedding of the maximal abelian cover of the complete graph on 4 vertices in 3-dimensional Euclidean space. It is also called the ‘triamond’, since it is a theoretically possible — but never yet seen — crystal structure for carbon.

# Diamond Cubic

This picture by Greg Egan shows the pattern of carbon atoms in a diamond, called the diamond cubic. Each atom is bonded to four neighbors. This pattern is found not just in carbon but also other elements in the same column of the periodic table: silicon, germanium, and tin.

# Togliatti Quintic

A quintic surface is one defined by a polynomial equation of degree 5. A nodal surface is one whose only singularities are ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$. A Togliatti surface is a quintic nodal surface with the largest possible number of ordinary double points, namely 31. Here Abdelaziz Nait Merzouk has drawn the real points of a Togliatti surface.

# Kummer Quartic

A quartic surface is one defined by a polynomial equation of degree 4. An ordinary double point is a point where a surface looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$. The Kummer surfaces are the quartic surfaces with the largest possible number of ordinary double points, namely 16. This picture by Abdelaziz Nait Merzouk shows the real points of a Kummer surface.

# Cayley’s Nodal Cubic Surface

A cubic surface is one defined by a polynomial equation of degree 3. Cayley’s nodal cubic surface, drawn above by Abdelaziz Nait Merzouk, is the cubic surface with the largest possible number of ordinary double points and no other singularities: 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 4 ordinary double points, shown here at the vertices of a regular tetrahedron.

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