Snowflake harmonics and computer graphics: Numerical computation of spectra on fractal drums

In this work, we study the steady-states vibrations of the ''Koch snowflake drum'', both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or ''frequencies'' of the drum), along with the corresponding eigenfunctions (called ''snowflake harmonics'') of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (''blow up'' or ''infinite stress'' of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions. According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and ''stabilization'' of fractal structures (e.g., coastlines, trees and blood vessels) in nature.