The Hofstadter butterfly is recast as a tessellation of trapezoids with integer slopes, governed by eight SL(2,Z) generators and connected to Farey, Apollonian, and Pythagorean structures, largely consolidating prior work.
, G.H.Wannier, phys. stat. sol. (b) 88, 767 (1978)
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The Hofstadter Butterfly: Bridging Condensed Matter, Topology, and Number Theory
The Hofstadter butterfly is recast as a tessellation of trapezoids with integer slopes, governed by eight SL(2,Z) generators and connected to Farey, Apollonian, and Pythagorean structures, largely consolidating prior work.