Togliatti, Labs, Endraß, Barth, Sarti — the surfaces holding the world records for singular points
How many sharp pinch points, called nodes, can a surface of degree d have? These spiky record holders are the best answers found so far, and from degree 7 up nobody knows the true limit.
Climb from Cayley’s 4-node cubic to Sarti’s 600-node surface. Drag to rotate, and raise the quality for high degrees so the fine spikes show.
Icosahedral symmetry and the golden ratio in every coefficient. 50 nodes are visible here; 15 more sit in the plane at infinity.
mu(6) = 65, proven maximal (Jaffe & Ruberman 1997)
A node is the mildest singular point there is: the surface pinches to a cone, locally looking like x² + y² = z². The node counts quoted are counts over the complex numbers, and only real points can be drawn — these models are chosen so that as many nodes as possible are real, but some (like 15 of the Barth sextic’s 65) sit in the plane at infinity, beyond any affine picture. What the pictures do show honestly is the symmetry: the Barth and Sarti surfaces wear their icosahedral five-fold symmetry openly.
A surface is smooth at a point if it has a well-defined tangent plane there — equivalently, if the gradient of the defining polynomial is not zero. Where the gradient vanishes, the surface is singular, and the mildest way that can happen is a node, also called an ordinary double point. Near a node the surface looks like the cone x² + y² = z²: two sheets touching at a single sharp point, like the tips of two ice-cream cones pressed together.
Nodes are the cheapest singularities, which is precisely why the competition is about them. Nastier singularities “use up” more of the surface’s budget, so if you want to cram in as many singular points as possible, plain nodes are what you want. The competition is a packing problem: how many pinch points can a polynomial of degree d support before it runs out of room?
One honesty note about the pictures. Node counts are counted over the complex numbers, but only real points can be drawn. Sometimes these agree — Barth deliberately built his sextic so that all 65 nodes are real. Sometimes they do not: push the Kummer family’s parameter below μ² = 1 in the demo below and twelve of its sixteen nodes go complex, leaving only four in the picture.
Compare the best construction with the proven ceiling, degree by degree. Left of the dashed line they meet; to the right, the shaded band is what nobody knows yet.
Hover a degree to read its story. Filled dots are settled records; hollow dots are merely the best anyone has managed so far, with the small amber dot marking the proven ceiling above them.
Write μ(d) for that maximum. Kummer settled degree 4 in 1864 with the answer 16. Togliatti found 31 nodes on a quintic in 1940. Barth built a sextic with 65 in 1996 and a degree-10 surface with 345 the same year. Sarti reached 600 in degree 12. And then the trail goes cold: from degree 7 upward, nobody knows the answer. There is a proven ceiling and a best-known construction, and a gap in between that has resisted everyone for decades.
Progress on μ(d) comes from two directions that have never quite met. From above, Miyaoka’s bound (1984) says a degree-d surface has at most (4/9)·d(d−1)² nodes, proved by techniques from the theory of surfaces of general type. For degree 8 that gives 174.2, so at most 174. From below, people build actual surfaces — and the best degree-8 construction, by Endraß in 1997, achieves 168. The gap between 168 and 174 has stood ever since.
The constructions are not random searches. Almost every record holder is built from symmetry: Barth’s sextic and decic and Sarti’s dodecic are invariants of the icosahedral group, which is why they look like spiky twenty-sided dice. Imposing a large symmetry group collapses the search space from millions of coefficients to a handful, and the nodes then arrive in whole orbits at a time. Labs found his 99-nodal septic in 2004 by an entirely different route: searching over finite fields, then lifting the answer back to characteristic zero.
The exact values are known only for degree 6 and below: μ(3) = 4, μ(4) = 16, μ(5) = 31, μ(6) = 65. Everything past that is an open problem in a subject where you can see the objects with your own eyes.
Push the parameter past μ² = 1 and watch a smooth quartic pinch: twelve nodes appear at once and drift apart.
Past μ² = 1 all sixteen nodes are real: at each yellow beacon two sheets meet at a single point, exactly like the tip of a cone. Twelve of them sit where the four tetrahedral planes cut each other; the other four were real all along. The full Kummer quartic has 16 nodes over the complex numbers, and here every one of them is visible in ordinary space.
Nodes are not accidents; they are what you get for free when a family of surfaces crosses a wall. The set of singular quartics forms a hypersurface — the discriminant — in the space of all quartics, and this slider drives a straight line through it. Each crossing is a moment where the surface must pinch, and building record surfaces means engineering a polynomial that crosses as deeply into the discriminant as possible.
Spot a mistake or have an idea?