I've been told time and time again that it is impossible to trisect an angle using a compass and a straightedge, but I just can't accept this.

Let us take a circle (a 360 degree angle). We then bisect the circle (as most of us learned in high school, bisecting an angle can be done with a straightedge and a compass relatively easily). Using our compass/straightedge combination we can find the midpoint of the diameter we drew earlier and draw a line perpendicular to the diameter (bisecting a 180 degree angle). We then have four 90 degree arcs of the circle, or four equal sections of our original angle. If we continue the process we increase the number of equal sections by doubling. We have 1 (the circle), 2, 4, 8, 16, 32, 64...

This sequence can be equated to 2

^{x}and if we want one third of the circle (trisected) this number must be a multiple of 3.

Therefore: 2

^{x}=3y

Since I know someone out there will be pedantic about this, x is restricted to positive integers.

Is it safe of me to assume that at some point 2

^{x}will be divisible by 3? Have I made any mistakes mathematically in this process?

If yes to the former and no to the later have I not just shown proof of concept that an angle can be trisected?