The “10 Martini” proof links quantum mechanics to infinitely complex mathematical structures

But in some respects, the evidence was a bit unsatisfactory. Zhitomirskaya and Avila used a method that only applies to certain irrational values ​​of alpha. By combining it with the intermediate evidence that preceded it, they can say that the problem has been solved. But this common proof was not elegant. It was a patchwork quilt, each square stitched from a different patchwork.

Moreover, the proofs only settled the conjecture as originally stated, which involved making simplifying assumptions about the electron’s environment. More realistic situations are more chaotic: the atoms in a solid are arranged in more complex patterns, and the magnetic fields are not completely static. “I’ve verified that for this model, but what does that have to do with reality?” said Simon Becker, a mathematician at the Swiss Federal Institute of Technology in Zurich.

These more realistic situations require modifying the part of the Schrödinger equation where alpha appears. When you do this, the 10 Martini Manual will stop working. “This was always annoying to me,” Zhitomirskaya said.

Analyzing the evidence in these broader contexts also suggests that the beautiful fractal patterns that emerged—the Cantor groups, the Hofstadter butterfly—were little more than a mathematical curiosity, something that would disappear once the equation became more realistic.

Avila and Zhitomirskaya moved on to other problems. Even Hofstadter had doubts. If one experience saw his butterflies, he would write Godel, Escher, Bach“I will be the most surprised person in the world.”

But in 2013, a group of physicists at Columbia University captured his moth in the laboratory. They placed two thin layers of graphene in a magnetic field, then measured the energy levels of the graphene’s electrons. The quantum fractal appeared in all its glory. “Suddenly it went from a figment of a mathematician’s imagination to something practical,” Zhitomirskaya said. “It has become very worrying.”