Euclid versus origami.

In 1936 Margherita Piazzolla Beloch, an Italian mathematician at the University of Ferrera, published a paper that proved that starting with a length L on a piece of paper, she could fold a length that was the cube root of L. She may not have realized it at the time, but this meant that origami could solve the problem given to the Greeks at Delos, where the Oracle demanded that the Athenians double the volume of a cube. [...] It also followed from Beloch's proof that any angle can be trisected--and this cracked the second great unsolvable problem of antiquity. Beloch's paper, however, remained in obscurity for decades, until in the 1970s the math world began to take origami seriously.

[...]

As it turns out, origami is more versatile than a ruler and compass, for example, in constructing the regular polygons. Euclid was able to draw an equilateral triangle, square, pentagon, and hexagon, but recall that the heptagon (which has seven sides) and the nonagon (nine) eluded him. Origami can fold heptagons and nonagons relatively easily, although it meets its mach with the 11-agon.

-Alex Bellos, Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math. New York: Free Press, 2010. (67)

This is a fine example of how deeply metaphors can be buried in human inquiry. The methods of geometry we learned in high school were mostly derived from Euclid, and the basic tools of Euclidean geometry assumed you could (1) draw connections between fixed points and (2) draw circles from two fixed points, using one as the center and letting the distance to the other provide the radius. Though by Euclid's time, this approach had been extended into three-dimensional space, it assumed a similar mathematical tool set and kept the idea of constructions against a fixed background.

Origami math adds folding, and thereby gains a really powerful tool. The mathematicians who followed the Greeks were stumped by three problems, two of which Bellos references above. But as a refresher, the problems were:

1. circle squaring: given a circle, use a compass and straightedge to construct a square with the same area
2. cube duplication: given a cube, use compass-straightedge methods to construct a cube with exactly twice the volume (note that since V = length * width * height, doubling the edges of the cube give you a second cube that is eight times as large)
3. angle trisection: given an angle, use a compass and straightedge to divide the angle into three equal parts (note that bisecting an angle is easy)

Had the earliest Greek mathematicians been interested in folding things rather than drawing on the ground or on walls, who knows what techniques they might have come up with? I'm not necessarily wishing that things had been otherwise; the efforts to solve these three problems -- and eventually to prove that no compass-straightedge solutions were possible -- led us to techniques far more powerful than folding. Perhaps having three problems that seem simple but are in fact unsolvable stokes curiosity in a particular way.

But on the other hand, it turns out that trisecting an angle is really, really simple when you're folding. It's like our civilization spent two thousand years looking for our keys or something.