Dividing Segments by Origami
By folding a square piece of paper of side length one, we can divide some of the edges of the square into rational lengths. This Demonstration shows two of the methods (known as Haga's first and third theorems) that focus on folding to touch the midpoint of the upper side of the square. By dragging the sliders to the right and back the two steps of each folding (step 1 is the same for both) can be performed in 3D. The final ratios can also be seen to lead to interesting constructions. For instance, the first folding shows how to trisect a segment into three parts using no tools! Also, we we can construct Pythagorean 3-4-5 triangles (that is, right triangles in which the ratio of their sides is in the proportion 3:4:5) with the same ease; see if you can spot them!
K. Haga, Origamics: Mathematical Explorations Through Paper Folding, River Edge, New Jersey: World Scientific, 2008.