Beyond compass and ruler

For most of the history of geometry, there were two tools: a compass and a straightedge. With them, the Greeks could bisect an angle, draw a perfect pentagon, and construct a great deal of the mathematics we still teach. But there were things they could not do, and three problems in particular resisted them for two thousand years.

Two of them — trisecting an angle, and doubling a cube — a folded sheet of paper can solve.

Why paper is stronger

A compass and straightedge can only solve equations up to the second degree. That is their ceiling, and in 1837 Pierre Wantzel proved that trisecting a general angle requires solving a cubic — a third-degree equation — which places it forever out of reach.

Folding paper does not have this ceiling. A single fold can be made to satisfy two conditions at once — bringing one point onto a line while bringing a second point onto a second line — and that simultaneous act is equivalent to solving a cubic. The rules of what a fold can do were eventually written down as the Huzita–Justin axioms, seven operations that define origami geometry the way compass and straightedge define classical geometry.

One caution, since we insist on them: paper cannot do everything. It cannot square the circle — that problem involves π, which no finite construction can reach. Two of the three ancient problems fall to a fold. The third does not.

The woman who saw it first

The credit for this discovery is tangled, and one name deserves rescuing. In 1936, the Italian mathematician Margherita Piazzolla Beloch showed that a single fold could solve a cubic equation — decades before the axioms were formally stated. She adapted a graphical method for solving equations, due to Eduard Lill, into an act of paper folding, and in doing so became the first person to see that origami reaches past the compass and ruler.

She is not in most textbooks. She should be.