Space-filling curves provide a graceful example of how mathematical objects can appeal to one's artistic sense. Similarly, Chladni visualization of a vibrating membrane renders artistically interesting plane curves, or nodal sets, of extreme mathematical value. We explore the nodal lines of particular eigenfunctions of the Dirichlet problem for the Laplace equation in a square domain, and we present a non-recursive family of simple curves which fills the space at the limit. This family of quasi-space filling curves provides a visually interesting application of the nodal set problem, which we also consider in the context of musical vibrations.