The square peg problem is at least 106 years old and still unsolved in full generality. It asks whether any simple closed curve in the plane inscribes a square. By “inscribes a square” we mean that the curve contains the four vertices of a square. The square itself may intersect both the bounded and unbounded components. Substantial progress on the problem has been made using methods from equivariant topology: Piecewise linear, analytic, convex, and locally monotone curves are all known to inscribe squares.
We present a recent positive result by T. Tao that takes an entirely different, “analytical” approach involving areas defined by line integrals and Stokes’ theorem.