Creation: Between Art and Mathematics

Creation: Between Art and Mathematics was a temporary exhibition at the Maison Poincaré (Institut Henri Poincaré) in Paris, on view April 9 to July 25, 2026 in the Espace Laurent Schwartz. Lines on a Cubic Surface (with Gabriel Dorfsman-Hopkins and Claudio Gómez-González), an interactive visualization of the twenty-seven lines on a cubic surface, was among the works shown.

The piece came out of a problem Gabe Dorfsman-Hopkins and I had been talking about for a while. Every complex cubic surface contains exactly twenty-seven lines. Over the reals the count can drop, but there is a moduli space of cubics that keep all twenty-seven of them real, and those arise from a uniform geometric construction: blow up the plane at six points. What we wanted was to make that construction something you could handle — six points you drag around a plane on one side of the screen, and on the other the cubic surface they determine, its twenty-seven lines swinging as you move them.

We tried two things first. One parameterized the surface directly from the blow-up, and the algebraic geometry of blow-ups made it too inaccurate. The other went after the implicit equation and ray-marched that, which meant Gröbner bases: we learned a great deal, and it was hopelessly slow for anything running in real time.

Then Claudio Gómez-González joined. Twenty-seven lines is an enormous amount of information to have about a surface, and his idea was to solve for the lines themselves first. Once you have them, linear algebra recovers the equation that contains them. I wrote the program that does it in shaders, and rendered the final image with a path tracer of my own. It took forty hours to converge.