  We describe a new method to construct Laplacians on fractals using a Peano
curve from the circle onto the fractal, extending an idea that has been used in
the case of certain Julia sets. The Peano curve allows us to visualize
eigenfunctions of the Laplacian by graphing the pullback to the circle. We
study in detail three fractals: the pentagasket, the octagasket and the magic
carpet. We also use the method for two nonfractal self-similar sets, the torus
and the equilateral triangle, obtaining appealing new visualizations of
eigenfunctions on the triangle. In contrast to the many familiar pictures of
approximations to standard Peano curves, that do no show self-intersections,
our descriptions of approximations to the Peano curves have self-intersections
that play a vital role in constructing graph approximations to the fractal with
explicit graph Laplacians that give the fractal Laplacian in the limit.
