But in some ways, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a method that only applied to certain irrational values of alpha. By combining it with an intermediate proof that came before it, they could say the problem was solved. But this combined proof wasn\u2019t elegant. It was a patchwork quilt, each square stitched out of distinct arguments.<\/p>\n
Moreover, the proofs only settled the conjecture as it was originally stated, which involved making simplifying assumptions about the electron\u2019s environment. More realistic situations are messier: Atoms in a solid are arranged in more complicated patterns, and magnetic fields aren\u2019t quite constant. \u201cYou\u2019ve verified it for this one model, but what does that have to do with reality?\u201d said Simon Becker<\/a>, a mathematician at the Swiss Federal Institute of Technology Zurich.<\/p>\n These more realistic situations require you to tweak the part of the Schr\u00f6dinger equation where alpha appears. And when you do, the ten martini proof stops working. \u201cThis was always disturbing to me,\u201d Jitomirskaya said.<\/p>\n The breakdown of the proof in these broader contexts also implied that the beautiful fractal patterns that had emerged \u2014 the Cantor sets, the Hofstadter butterfly \u2014 were nothing more than a mathematical curiosity, something that would disappear once the equation was made more realistic.<\/p>\n Avila and Jitomirskaya moved on to other problems. Even Hofstadter had doubts. If an experiment ever saw his butterfly, he\u2019d written in G\u00f6del, Escher, Bach, \u201cI would be the most surprised person in the world.\u201d<\/p>\n