Spectra of almost periodic matrices from Oliver Knill on Vimeo.

Spectra of almost periodic matrices in 4K resolution. The matrices (found in 2008 for a talk at Dunster house and used in 2010 for a Mathematica project for a Math 21b course) have the entries A(n,m) = cos(n m a + n b), where a,b are some fixed real parameters. The movie shows what happens, when the parameters a,b are slowly changed. The matrices are 2160 x 2160 matrices. The pictures show the spectra (the eigenvalues of the matrix in the complex plane). They were exported to 3840x2160 pixel pictures from the computer algebra system Mathematica. About 7800 frames were computed starting from a=sqrt(2) and b=sqrt(5) increasing linearly (very slowly!) The color encodes how close the nearest eigenvalue is. The spectra always look similar. There is an exact symmetry for complex conjugation as the matrices are real. The symmetry with respect to the y axes is a statistical phenomenon. The fractal nature of the spectra is unexplained. Almost nothing is known yet.