How can you trisect an angle? It can be shown it's impossible to do this with ruler and compass alone, (using Galois theory) - so don't try it!!! But you may be able to find some good approximations. However, in origami, you can get accurate trisection of an acute angle.
You can read about this in several places, but since it's so neat, I thought I'd put instructions up here too - more people should be able to do this for a party trick!
Jim Loy has informed me that this construction is due to to Hisashi Abe in 1980, (see "Geometric Constructions" by George E. Martin). See Jim Loy's page at http://www.jimloy.com/geometry/trisect.htm for a description of many other ways to trisec an angle.
Since we're working with origami, the angle is in a piece of paper:
So what we want is to find how to fold along these dotted lines:
Note, if you don't start with a square, you can always make a square, here's the idea.
We're going to trisect this angle by folding. I'm going to try and describe this in a way so that you'll remember what to do.
Suppose we could put three congruent triangles in the picture as shown:
These triangles trisect the angle. So we need to know how to get them there.
Choose some height for the lower triangle, any height, and crease a horizontal line at this height; ie, just crease any horizontal line you want:
We need to get the blue line of the following picture somehow:
We can make a kind of "marked ruler" in the side of the paper, by folding over the paper again:
Now this "marked ruler" is used to find the bold line we needed:
To do this, fold the paper so point b touches line B, and point d touches line D:
You can check that this really does work out, and that the angles are the same for these triangles:
(Note, in the above, we don't really have a marked ruler as such, as we can't move the edge of the paper to any position, as it's attached to the rest of the paper.)
You can find another accounts of this construction at: