Continuum of 6-Colorings of the Plane

In 1993, another 6-coloring was found by Ilya Hoffman and me ([HS1], [HS2]). Its type was 1 1 1 1 1 2 − 1 . $$ \left(1,1,1,1,1,\sqrt{2}-1\right). $$ The story of this discovery is noteworthy. In the summer of 1993, I was visiting my cousin in Moscow, a well-known New Vienna School composer, Leonid Hoffman. His 15-year-old son Ilya was studying violin at the Gnessin Music High School. Ilya set out to learn what I was doing in mathematics and did not accept any general answers. He wanted particulars. I showed him my 6-coloring of the plane (Problem 6.4 ), and the teenage musician got busy. The very next day he showed me … the Stechkin coloring (Fig. 6.2 ) that he discovered on his own! “Great,” I replied, “but you are 23 years late.” A few days later, he came up with a new idea of using a two-square tiling. Ilya had an intuition of a virtuoso fiddler and no mathematical culture – and so I calculated the sizes the squares had to have for the 6-coloring to do the job we needed. I wanted Ilya to be the sole author, but he insisted on our joint credit. And the joint work of the unusual mathematician–musician team was born. Ilya went on to graduate from the school of Moscow Conservatory in the class of the celebrated violist and conductor Yuri Bashmet and is now one of Russia’s hottest violinists and violists and the winner of several international competitions.