# 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 \( x^2 + y^2 = z^2\).

# Barth Decic

A **decic surface** is one defined by a polynomial equation of degree 6. The **Barth decic**, drawn here by Abdelaziz Nait Merzouk, is the decic 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 \(x^2 + y^2 = z^2 \).

# Small Stellated Dodecahedron

The **small stellated dodecahedron**, drawn here using Robert Webb’s Stella software, is made of 12 **pentagrams**, or 5-pointed stars, with 5 pentagrams meeting at each vertex.

# Discriminant of Restricted Quintic

This image by Greg Egan shows the set of points \((a,b,c)\) for which the quintic \(x^5 + ax^4 + bx^2 + c \) has repeated roots. The plane \(c = 0 \) has been removed. This surface is connected to involutes of a cubical parabola and the discriminant of the icosahedral group.

# 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.

# Involutes of a Cubical Parabola

This animation by Marshall Hampton shows the involutes of the curve \(y = x^3\). It lies at a fascinating mathematical crossroads, which we shall explore in a series of three posts.