Let me start right away by shouting out loud: The trisection of an angle with compass and ruler is impossible!!!
Okay, after this disclaimer we can start discussing the problem. The proof that the trisection is indeed not possible for general angles is due to Wantzel in 1836 (according to Wolfram Math World). It is based on the fact that the trisection would solve a very general rational polynomial equation of degree 3, but such a solution can in general not be written in terms of square roots, which is all a compass and a ruler can achieve. Note that some angles can be trisected, some can’t. The result does not only claim, that there is no general method. It claims that there are angles (indeed most in some sense) that can not be trisected, not by any compass and ruler construction.
There are sliding constructions, which are not really constructions. See this C.a.R. page for two examples. These „constructions“ slide a line such that two requirements are solved simultaneously. Each of these can be solved by a compass and ruler construction, but not both at the same time. This is not cheating. It is just not the way the ancient Greek thought about constructions. In view of Wantzel’s result it is like allowing cube roots in equations, where previously only square roots were allowed.
There are good approximations. One is shown on Wofram’s site. But I like another one better, because it so good.
- Trisect the chord segment of the angle, and intersect the radius through this point with the circle for a point A.
- Construct a line through the trisection point of the chord segment perpendicular to the segment, and intersect it with the circle for a point B.
- Trisect AB to find a point C.
The error is of order 5, and not larger than 0.007° for angles less than 45°. A and B are already good approximations with an error less than 0.36° resp. 0.17°. To be fair, it is very hard to trisect AB in a real world construction, since it is already very small.
The angle trisection is only one of the Greek problems. Another one is to construct a square with the same area than a given circle. This is even harder, since pi is not a zero of any rational polynomial. It is transcendental.
If you like some fun read Dudley’s What to do when the trisector comes.
But, can a „equation of degree 3“ be written in terms of square roots using non-euclidean spaces which can be constructed using square roots?