Cubic equations and the ideal trisection of the arbitrary angle

摘要

In the year 1837 mathematical proof was set forth authoritatively stating that it is impossible to trisect an arbitrary angle with a compass and an unmarked straightedge in the classical sense. The famous proof depends on an incompatible cubic equation having the cosine of an angle of 60° and the cube of the cosine of one-third of an angle of 60° as parameters. This article re-examines the cubic equations linked with the trisection of an arbitrary angle and presents evidence showing that, where the arbitrary angle is less than or equal to the maximum central angle of 180°, a cubic equation having the cube of the sine of one-third of half of any central angle as a parameter consistently is compatible with the ideal trisection of the central angle. Amazingly, the sine of one-third of half of any central angle minus one-third of the sine of half of the central angle (a rational operation) consistently is equal to four-thirds times the cube of the sine of one-third of half of the central angle. In view of the consistently compatible cubic equations and other findings presented, perhaps the long-standing proof of impossibility should be re-examined.