A sculptural gallery of the du Val singularities — and the Dynkin diagrams hiding inside each one
Some singularities are wild. These are the tame ones — the simple surface singularities, also called du Val singularities or rational double points. There are exactly five families of them, and their names will look familiar to anyone who has met a Lie algebra: An, Dn, E6, E7, E8.
That is the first surprise. The second is why. Take any of these surfaces, resolve the singular point by blowing it up until the surface is smooth, and look at what you replaced the point with: a little constellation of spheres, each meeting some of the others. Draw a dot for every sphere and a line for every crossing, and you have drawn a Dynkin diagram — the very same picture that classifies the simple Lie algebras. The geometry does not merely resemble the algebra; it reproduces it exactly, entry for entry.
And then the third surprise, which is the one that makes people sit up: the same list classifies the finite groups of rotations in SU(2), which is to say the symmetry groups of the Platonic solids. E8, the largest exceptional diagram in mathematics, is the icosahedron in disguise.
The nine simple surface singularities, extracted by marching cubes and lit like museum pieces. Start with A1 — the ordinary double cone, two sheets pinched to a single point — and walk up through the chains, the forks, and the three exceptional sculptures. The pulsing beacon marks the singular point itself.
The ordinary double point: two smooth sheets meeting at a single point. One blow-up resolves it.
The ordinary double cone — the only du Val singularity whose real picture equals the textbook picture.
These sculptures are real slices of complex objects: the honest du Val singularities live in C³, where all sign choices are equivalent. Over the reals we pick the sign variant with the richest picture — the pulsing beacon marks the singular point at the origin.
A surface singularity is a point where the surface fails to be smooth — where it pinches, crosses itself, or comes to a cone. The du Val singularities are the ones that are as mild as a genuine singularity can be. Formally they are the rational double points: resolving them costs nothing in terms of the surface’s global invariants, because the exceptional curves you insert are all spheres of self-intersection −2.
Every one of them arises the same way. Take the complex plane C2 and a finite group Γ of rotations acting on it — a finite subgroup of SU(2). The quotient C2/Γ is a surface with exactly one singular point, sitting where the origin was, and every simple surface singularity is one of these quotients. The classification of the singularities is the classification of the finite subgroups of SU(2). This is the McKay correspondence.
The sculptures above are honest but partial: they are the real slices of complex surfaces. Over C the sign choices in x² + y² + zn+1 = 0 do not matter, since you can absorb them by scaling coordinates. Over R they matter a great deal — the all-plus forms often have almost no real points at all — so each sculpture uses the sign variant with the richest real picture. What you are looking at is a shadow of the true object, chosen to be the most photogenic shadow available.
Blow up the singular point and watch the exceptional spheres appear one step at a time. When the surface is finally smooth, the spheres you inserted form the Dynkin diagram. Toggle the matrix to see the punchline: the intersection numbers of those spheres are precisely the negative of the Cartan matrix.
x² + y² + z⁵ = 0
One singular point. Nothing resolved yet — press a blow-up.
| -2 | 1 | 0 | 0 |
| 1 | -2 | 1 | 0 |
| 0 | 1 | -2 | 1 |
| 0 | 0 | 1 | -2 |
Every diagonal entry is −2: each exceptional curve is a sphere of self-intersection −2. Every off-diagonal 1 marks a point where two spheres cross. The two matrices are negatives of each other — the geometry of the resolution and the algebra of the Lie group A₄ are literally the same table of numbers.
A blow-up replaces a point by the set of all directions through it — a whole projective line, P1, which over the complex numbers is a sphere. It is a surgical tool: away from the point nothing changes, but the pinch at the centre is pulled apart and spread over the new sphere. Do it enough times and the surface becomes smooth. The union of the spheres you inserted is the exceptional divisor, and its dual graph — one node per sphere, one edge per intersection — is the resolution graph.
For An the graph is a chain of n spheres. For Dn the chain sprouts a two-pronged fork at one end. For E6, E7, E8 you get the three exceptional trees. No other graphs occur. That is the entire classification, and it is forced: the intersection form has to be negative definite, and the negative-definite simply-laced graphs are exactly the ADE diagrams.
You can explore blow-ups in their own right in the Algebraic Geometry module’s Singularities & Blow-ups lesson, and meet Dynkin diagrams on their home turf in Lie Groups.
Four columns, one classification. Click any row to light up a Platonic solid, a finite group of rotations, a Dynkin diagram, and a singular surface — four objects from four different parts of mathematics that turn out to be the same object wearing different clothes.
| Symmetric solid | Subgroup of SU(2) | Dynkin diagram | Singularity |
|---|---|---|---|
Cyclic rotations of an n+1-gon | cyclic ℤ/(n+1) order n + 1 | Aₙ | x² + y² + zⁿ⁺¹ = 0 |
Dihedral symmetries of a prism | binary dihedral 2D(n−2) order 4(n − 2) | Dₙ | x² + y²z + zⁿ⁻¹ = 0 |
Tetrahedron | binary tetrahedral 2T order 24 | E₆ | x² + y³ + z⁴ = 0 |
Octahedron / cube | binary octahedral 2O order 48 | E₇ | x² + y³ + yz³ = 0 |
Icosahedron / dodecahedron | binary icosahedral 2I order 120 | E₈ | x² + y³ + z⁵ = 0 |
The largest exceptional. The icosahedron (and its dual dodecahedron) has rotation group of order 60 — the alternating group A₅ — whose double cover in SU(2) has order 120. Klein showed in 1884 that its quotient is x² + y³ + z⁵ = 0, and its resolution graph is E₈, the biggest exceptional Dynkin diagram.
The same A-D-E list turns up in yet more places: the simply-laced simple Lie algebras, the quivers of finite representation type (Gabriel’s theorem), the finite subgroups of SU(2), and the simple critical points of functions. Nobody considers this a coincidence any more — but a single reason covering every case is still elusive.