In two-dimensional space, a curve can be cut out by a single equation: A line might be written as y = 3 x − 7, a circle by x 2 + y 2 = 1. “This is really about just understanding what curves are,” said Ravi Vakil, a mathematician at Stanford University.īut curves that live in higher dimensions, despite having been studied with state-of-the-art tools for hundreds of years, are tricky beasts. It’s a fundamental question - known as the interpolation problem - about algebraic curves, one of the most central objects in mathematics. More generally, mathematicians want to know when you can draw a curve through arbitrarily many points in arbitrarily many dimensions.

But you can pass a circle through any three points, and a conic section (an ellipse, parabola or hyperbola) through any five. Any more points, and you’re out of luck: It’s not likely that a single line will pass through all of them. A basic fact of geometry, known for millennia, is that you can draw a line through any two points in the plane.

Pure point measure