Lectures on the Philosophy of Mathematics

by Joel David Hamkins

Robert Geretschlager (1995) explains the equivalence of straightedge-and-compass construction with an axiomatic list of seven fundamental origami folds, making reductions in both directions. He also proves, however, that certain more complex origami folds go strictly beyond Euclidean constructibility. One kind of fold, for example, can be described as allowing one to fold along the common tangent to two parabolas, as specified by given focal points and vertex lines (one can in effect achieve this with actual paper). With the addition of this folding construction, it turns out, one can solve arbitrary cubic equations over the rational numbers, constructing and cube roots generally, which are not all constructible by straightedge and compass, as discussed previously. One can also trisect any given angle using these origami constructions. Thus, origami is realized ultimately to have a mathematical power surpassing the Euclidean tool set.