What's the closest anyone has got to trisecting an angle with compass and straghtedge?

Question

I know it's not possible to perfectly, trisect an angle with compass and straightedge, but what's the closest anyone has gotten to doing this?

Answer 1

It's easy to "quadrisect" an arbitrary angle by bisecting it twice. You can get arbitrarily close to trisecting the original angle by repeatedly quadrisecting the angle and adding together the fractions that you generate, since $$ \frac13=\frac14 + \frac14\cdot\frac14 +\frac14\cdot\frac14\cdot\frac14+\cdots. $$

Answer 2

I dug up the following method from the 1966 Los Angeles Times:

Both methods work, but I'd say the one at the top is best.