For those of us who are, or were, avid paper-folders, there’s a flourishing field of mathematically-inspired origami. This goes in two directions: using origami to illustrate mathematical principles or theorems, and discovering new mathematical and algorithmic aspects to origami.
It’s pretty easy to see how this works, the paper is first folded in half both vertically and diagonally, and then a fold is made from the middle of the top edge to the bottom corner. A fold over the point where this last fold and diagonal fold meet is a third of the square.
The angle to be trisected is at the bottom left of the first figure. Start with making two folds equally spaced (this can be done by making any horizontal fold, and then folding the bottom edge up to the fold just made). Second, fold the bottom corner as shown so that the points coincide with the lines of the folds. A line from the bottom left corner to the middle point will trisect the angle. The difference between this construction and ruler-and-compass constructions is that in the latter we are not permitted to line up points with lines as we did here.
There’s a vast amount of other material out there – go and find it!