  Consider a family of bounded domains $\Omega_{t}$ in the plane (or more
generally any Euclidean space) that depend analytically on the parameter $t$,
and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then
we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$
with eigenvalues $\lambda_{t}^{(j)}$ also varying continuously with $t$,
although the relative sizes of the eigenvalues will change with $t$ at
crossings where $\lambda_{t}^{(j)}=\lambda_{t}^{(k)}$. We call these families
homotopies of eigenfunctions. We study two explicit examples. The first example
has $\Omega_{0}$ equal to a square and $\Omega_{1}$ equal to a circle; in both
cases the eigenfunctions are known explicitly, so our homotopies connect these
two explicit families. In the second example we approximate the Sierpinski
carpet starting with a square, and we continuously delete subsquares of varying
sizes. (Data available in full at www.math.cornell.edu/~smh82)
