# Math 21a Summer 2020

## Multivariable Calculus

# Square Peg

The**square peg problem**asks whether every simple closed curve in the plane admits four points which form a perfect square. The problem has been asked in 1911 by Otto Toeplitz. The problem has recently been solved for smooth curves. For general continuous curves, the case remains open.

A simple closed curve is a continuous curve which has no self intersection. For us in calculus, this means that we have two functions x(t), y(t) which are continuous and which are periodic in the sense x(t+1)=x(t), y(t+1)=y(t) and such that if [x(t),y(t)] = [x(s),y(s)] then t-s is an integer. Such curves can be very complicated.

In the case of continuous curves the square peg problem is still an open problem. For piecewise real analytic curves, the problem has been solved since more than 100 years. For smooth curves, the existence of rectangles is known to exist since 1944. See the Wikipedia article.

Here is the Quanta Magazine article reporting on a solution for the smooth case, posted in May 2020.