# Where can I get an introduction to the mathematics behind Hofstadter's Butterfly?

Are there any good books that give good mathematical/physical background to the workings of the Hofstadter's Butterfly? I'd appreciate some references. Books or Public access papers will work.

Preferably, I'd like something that gives some insight to things like the alteration of the Hamiltonian for magnetic fields among other things not covered in introductory QM course. Also, I'm interested in the numerical procedure used to actually plot the Butterfly. I'm willing to learn though, so if you post a prerequisite list, that'd work too.

• I'm curious why this post was made "community wiki". Is it because it's a resource recommendation question? I don't see anything about that in the linked policy. Mar 23 '16 at 14:34
• No reason really, questions just seemed to be better received this way. Mar 23 '16 at 16:41

I also recommend this pithy half-page summary of the math by Oliver Knill, that was linked to by the page that @Jitter posted a link to. It describes a function $f(x,y)$ for the Butterfly, using a Lyapunov exponent. Prerequisite: knowledge of matrices and determinants.