I found this on sci.math and thought this group might be interested: (Original posted by Sergei Markelov firstname.lastname@example.org)
*** It is well-known, that some angles (for instance, Pi/3) cannot be trisected using the ruler and compass. However, some angles still can be triseced. But which angles can be, and which cannot be trisected? I have found no answer in literature.
My ideas are:
1. If cos(Alpha) is transcendental, then construction is impossible. 2. If Alpha/Pi is rational, Alpha=p/q*Pi, then Alpha cannot be trisected if q=3k and can be trisected if q=3k+1 or q=3k+2 (see example with Pi/7 below). 3. If cos(Alpha) is algebraic, but Alpha/Pi is irrational - I have samples where Alpha can be triseced, and where Alpha cannot be trisected (artan(11/2) can be trisected, arctan(1/2) cannot, see below).
I have the proof of (1), have some ideas about how to prove (2), and no ideas about how to determine, whether the angle can be trisected in the case (3).
Here are few examples of angles that can be triseced, but this is not obvious:
1. Pi/7 can be trisected since Pi/21 = Pi/3 - 2*Pi/7 2. arctan(11/2) can be trisected since arctan(11/2) / 3 = arctan(1/2) 3. arctan((1+3*2^(1/3))/5) can be trisected since arctan((1+3*2^(1/3))/5) = 3*arctan(2^(1/3)-1)
Both these formulas can be proved using
tan(3*arctan(x)) = (3*x-x^3)/(1-3*x^2)
However, arctan(1/2) cannot be trisected, because minimal polynom for tan(arctan(1/2)/3) is 2*x^3-3*x^2-6*x+1 and it is irreducible over Q.
Both arctan(1/2) and arctan(11/2) are incommensurably with Pi (I have some ideas, how to prove this), but first cannot be trisected, and second can be...