# Chmutov Octic

An octic surface is one defined by a polynomial equation of degree 8. This image by Abdelaziz Nait Merzouk shows an octic discovered by Chmutov with 154 real ordinary double points or nodes: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$.

# Romik’s Ambidextrous Sofa

The ambidextrous moving sofa problem is to find the planar shape of maximal area that can negotiate right-angled turns both to the right and to the left in a hallway of width 1. The current best known solution was found by Dan Romik, and is shown here.

The Bunimovich stadium is a rectangle capped by semicircles in which a point particle moves at constant speed along straight lines, reflecting off the boundary in a way that the angle of incidence equals the angle of reflection. This animation, made by Phillipe Roux, shows a collection of such particles initially moving in the same…

# Escudero Nonic

A nonic surface is one defined by a polynomial equation of degree 9. This image by Juan García Escudero shows a nonic surface called $Q_9$, which has 220 real ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$.

# 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.

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