David S. Richeson in Lapham’s Quarterly:
The four impossible “problems of antiquity”—trisecting an angle, doubling the cube, constructing every regular polygon, and squaring the circle—are catnip for mathematical cranks. Every mathematician who has email has received letters from crackpots claiming to have solved these problems. They are so elementary to state that nonmathematicians are unable to resist. Unfortunately, some think they have succeeded—and refuse to listen to arguments that they are wrong.
Mathematics is not unique in drawing out charlatans and kooks, of course. Physicists have their perpetual-motion inventors, historians their Holocaust deniers, physicians their homeopathic medicine proponents, public health officials their anti-vaccinators, and so on. We have had hundreds of years of alchemists, flat earthers, seekers of the elixir of life, proponents of ESP, and conspiracy theorists who have doubted the moon landing and questioned the assassination of John F. Kennedy.
Circle squarers and angle trisectors have been around for as long as the problems themselves. The ancient Greeks used the word τετραγωνιζειν (tetragonidzein), which translates “to occupy oneself with the quadrature,” to describe those trying to solve the circle-squaring problem.