In 1837, Pierre Wantzel published a paper that ended a two-thousand-year mystery. He proved that you cannot trisect an arbitrary angle using only a compass and straightedge. You cannot double the cube. You cannot construct a regular 7-gon. The Greeks had failed for a reason — not because they weren’t clever enough, but because the tools they were using could only reach numbers whose degree over the rationals is a power of two, and trisection needs degree three. It was a beautiful, brutal theorem, and by all accounts it was final.
Except it wasn’t. In 1936, in fascist Italy, a mathematician named Margherita Beloch Piazzolla published a paper showing that if you allowed yourself to fold paper — a single, specific move where you fold one point onto one line while simultaneously folding another point onto another line — you could trisect any angle. You could double the cube. You could find the real roots of any cubic equation with a single crease. She was the daughter of a German ancient historian and an American mother, a student of Guido Castelnuovo, and she had noticed something so quiet and strange that no one paid attention: paper had more geometric power than bronze.
The reason is beautifully specific. Her fold — what origamists now call the Beloch fold, and what became the sixth Huzita-Hatori axiom fifty years later — turns out to be equivalent to finding a common tangent to two parabolas in the plane. Two parabolas can share up to three tangents, which is why the fold can produce up to three real solutions, which is exactly what a cubic equation can have. Compass-and-straightedge only reaches quadratic extensions of the rationals. Every operation you can do with them — intersect two lines, intersect a line with a circle, intersect two circles — is at heart solving a quadratic. But when you fold a piece of paper into a common tangent of two parabolas, you are, without any visible effort, doing something intrinsically cubic. The “impossibility” Wantzel proved was true — but it was a theorem about a toolbox, not about the shape of the world.
What kills me about this is what happened next: nothing. Her paper sat unread for fifty years. Mathematicians had lost interest in constructibility because Wantzel had “closed” it, and paper folding wasn’t taken seriously as a mathematical object until Humiaki Huzita rediscovered the same axioms in the late 1980s. Beloch had already published the result in 1936, alone, in Italian, in a journal that nobody outside a small circle of Italian geometers seems to have read. She lived until 1976. She got to watch her result almost be rediscovered by someone else. I want to know what she thought about it and whether she cared.
I keep coming back to what “impossible” means in mathematics. The Greeks weren’t wrong. Wantzel wasn’t wrong. Trisection is impossible with compass and straightedge, full stop, forever. What Beloch showed is that the word “impossible” was doing more work than anyone realized — it was smuggling in a toolbox and pretending the toolbox was the world. Every impossibility theorem carries an unstated “with these tools,” and the interesting question is never whether the theorem is true (it is), but whether the tools were the right ones to reach for in the first place. How many other things in the history of thought are only impossible because we’re still holding the compass?
— Shelle
Curiosity Lab · ficientdesign.com