The ‘10 Martini’ Proof Connects Quantum Mechanics With Infinitely Intricate Mathematical Structures
But in some ways the evidence was a little unsatisfactory. Zhitomirskaya and Avila had used a method that only applied to certain irrational values of alpha. By combining it with intermediate evidence that preceded it, they could say that the problem was solved. But this combined proof was not elegant. It was a patchwork, each square stitched with distinct arguments.
Moreover, the evidence only confirmed the conjecture as originally formulated, which involved making simplifying assumptions about the electron’s environment. More realistic situations are more complicated: the atoms of a solid are arranged in more complex patterns and the magnetic fields are not completely constant. “You checked it for this model, but what does this 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. And when you do, the 10 proof martini stops working. “It always bothered me,” Zhitomirskaya said.
The failure of the proof in these broader contexts also implied that the magnificent fractal patterns that had emerged—Cantor’s sets, Hofstadter’s butterfly—were nothing more than a mathematical curiosity, something that would disappear once the equation was made more realistic.
Avila and Zhitomirskaya moved on to other issues. Even Hofstadter had doubts. If ever an experience had allowed him to see his butterfly, he would have written Godel, Escher, Bach“I would be the most surprised person in the world.”
But in 2013, a group of physicists from Columbia University captured his butterfly in a 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 has appeared in all its splendor. “Suddenly it went from a figment of the mathematician’s imagination to something practical,” Zhitomirskaya said. “It’s become very disturbing.”
