Fill a large matrix with pure noise and its eigenvalues do something no one has any right to expect: they settle into a precise semicircle, and they refuse to sit close to one another. In 1972, over tea at Princeton, a number theorist described the spacing of the zeros of the Riemann zeta function to a physicist, who recognised it instantly as the spacing of those same eigenvalues — the statistics of heavy atomic nuclei. Across 40 interactive demonstrations, generate the matrices, compute the zeros, and watch two calculations with nothing in common land on the same curve. Nobody knows why.
See also: Number Theory for the zeta function itself, Linear Algebra for what an eigenvalue is, and Probability for the limit theorems this subject quietly replaces.
Fill a matrix with noise and its eigenvalues arrange themselves into a semicircle
Eigenvalues refuse to sit close together, and random points have no such manners
Three symmetry classes, three exponents, and nothing else in between
Too complicated to solve, so he replaced the physics with dice — and it worked
The largest eigenvalue has its own universal law, and it is not a bell curve
Change what you put in the matrix and the local statistics do not notice
The zeros of the zeta function are spaced like the energy levels of a heavy nucleus
Covariance matrices, the noise floor of PCA, and eigenvalues that fill a disc
Shuffle a deck or grow a crystal and the same distribution turns up again
Fifty years on, nobody can say why prime numbers and nuclei share a law