Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.  

In the following, *trisectable* always means with straightedge and compass alone.


I could however find nothing but some vague statements about angles, that are *trisectable*; a typical such statement is, that up to a few exceptions, it is impossible to trisect angles.


**Question:** I would therefore like to know, whether the angles, that are *trisectable* have been fully characterized and, if yes, who has done that first. 


**Edit:**  

in view of the valuable feedback I got, I will try to summarize things from my perspective


* it can be proven, that an angle $\theta$ can be constructed if and only if $\cos(\theta) \in \mathbb{E},$ where $\mathbb E$ (for Euclidean), often called the "constructable numbers," is the smallest subfield of the real numbers that is closed under taking square roots of positive elements.
* there seems to be common agreement, that an angle $\theta$ can only be trisected with straightedge and compass, if $\cos(\frac{\theta}{3}))$ is constructable.


So either being trisectable and constructable are equivalent or, there are angles that are not constructable, but can be trisected.


I suspect however, that there are certain angles (e.g. $\frac{2\pi}{7+\frac{1}{3}}$), that can be trisected despite not being constructable, namely angles $\theta$, for which


$$ \frac{lcm(\theta,2\pi)}{\theta} = 3k, k\in\mathbb{N}$$ 


those angles, which I would like to call *auto-trisecting*, are then either all constructable or, there are exceptions which are counter examples to the characterisation via constructability (the notion of being *auto-trisecting* of course easily generalizes to being *auto-$n$-secting*).


Interestingly, the only "basic" (i.e. involving only a single Fermat prime) constructable angles, that can be trisected, but are not *auto-trisecting*, seem to be the ones of the form$$3k\frac{2\pi}{2^nF\_0}, k\in\mathbb{N},n\in\mathbb{N}\_0, F\_0 := 2^{2^0}+1=3$$  

With that observation in mind, and assuming that all trisectable angles are either constructable or *auto-trisecting*, a bullet-proof strategy for finding a trisection of an angle, that is known to be trisectable with straightedge and compass alone, is to simply to construct multiples of it, until a period has been completed. The so generated multiples either 


* subdivide the angle in case it is *auto-trisecting* and the size of a trisection is equal to the sum of $\frac{1}{3}$ of the (equal) subdivisions or,
* do not subdivide the angle, but constitute to a constructable regular n-gon and, by determining n, it is possible to determine a constructable angle that resembles a trisection.


in both cases it is possible to find a trisection and, alltogether it can be said that the topic of trisecting an angle in case that its trisectability is known, can be discussed at classroom-level.


As a remark it can be said, that there is no hope for proving the existence of further Fermat primes by demonstrating that the known Fermat primes together with *auto-trisecting* angles do not cover all cases of trisectable angles and thus would necessitate the existence of further Fermat primes.


**Edit II:**  

I found the following Wiki article that elucidates the history of proving that angles are not trisectable in general:  

<http://en.wikipedia.org/wiki/Angle_trisection>  

according to that article, the proof was found by Pierre Wantzel in 1837 and is not based on Lindemann's proof of the transcendence of $\pi$, but rather on Galois theory. 



Characterization of Angles Trisectable with Straightedge and Compass
There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure for beginning with $\theta$ and using a finite number of compass and straightedge operations to actually find $\theta/3,$ even when it is constructable. 


I wrote an article on the hyperbolic plane, constant curvature $-1.$ As pointed out by Bolyai, there are a countably infinite set of pairs, one circle and one square, such that the radius of the circle and the edge of the square are both constructable, and the two figures have the same area. However, there is no way to begin with one and find the other. The phrase "square the circle" is misleading, the reverse phrase "circling the square" would also be misleading if anyone ever used it. 


I need to backtrack a little on that last phrase. If you have some sort of political plot going on in Moscow, Russia, and the hero is going around the perimeter of Red Square looking for the bad guys, it makes a good deal of sense to say that the hero is circling the square. The same would work with Times Square in New York.


Back to the Euclidean plane. Given an angle $\theta $ but **not told what it is** (in radians, say) there is no procedure for finding $\theta/3.$ Ever.


