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.
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.
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.
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 \( x^2 + y^2 = z^2\).
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 \(x^2 + y^2 = z^2 \).
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.
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.
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.
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.
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 \(x^2 + y^2 = z^2\).
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.
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.
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.
This animation by Greg Egan shows 27 lines on a surface defined by cubic equations: the Clebsch surface. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.
This is the Hoffman–Singleton graph, a remarkably symmetrical graph with 50 vertices and 175 edges. There is a beautiful way to construct the Hoffman–Singleton graph by connecting 5 pentagons to 5 pentagrams.
